mm-95 === How can one show for x, y in lp space with |x|,|y|=1 that |(x+y)/2|=1 => x=y. > I > am assuming here that> 1= 0 and for x >= a, de?e f(x) = (a^p + x^p)/2 - [(a+x)/2]^p.Then f'(x) > 0 if x > a. === > Here is an interesting question that I came across> given N points on a two dimensional plane, one can de?e a distance > matrix A = [a ] of order N x N such that a is the distance between> ij ij> the point i, and j.> The question is given this matrix can a set of points be given that> will satisfy the given distance matrix in an ef?ient manner?> abiA triangle given 3 sides satisfying triangle inequality in Euclidean2D can be constructed upto one arbitrary re?n about any axis inthe 2D and one arbitrary rotation, as is well known. The process canstart by taking ?st point at origin and second point to be, say onx-axis.To determine a new point with respect to 3 given points in thetriangulation process ( similar to civil engineering surveying ), weneed 3 distances from them to the new point. Accordingly, ifN number of points are given, p = 3 N-6 distances are needed exactlyfor uniquely determining their position/embedding in 2D. If more thanp distances are given, then there is inconsistency/redundancy. Sowhen these total p combination distances are given, the position ofall points are intrinsically ?ed in relation to each other, exceptfor the arbitrariness involved in ?st three points mentioned above.HTH === > Here is an interesting question that I came across> given N points on a two dimensional plane, one can de?e a distance > matrix A = [a ] of order N x N such that a is the distance between> ij ij> the point i, and j.> The question is given this matrix can a set of points be given that> will satisfy the given distance matrix in an ef?ient manner?> abiThe book by Deza and Laurent ?Geometry of cuts and metrics' is a goodreference in this regard. For the speci? problem you ask, I believethere is a theorem of Menger (I don't have the book handy) that statesthat you need to check that the generalization of the triangleinequality (and the tetrahedral inequality that Dave Rusin mentioned)is true for all subsets of d+3 points holds (in this case 5 points).The condition is necessary and suf?ient.Of course being a metric is a necessary condition for any of the aboveto work. === > given N points on a two dimensional plane, one can de?e a distance > matrix A = [a ] of order N x N such that a is the distance between> ij ij> the point i, and j.> The question is given this matrix can a set of points be given that> will satisfy the given distance matrix in an ef?ient manner?Let C = (-1/2)(I - uu'/n)B(I - uu'/n), whereB be the matrix of squared distances among the n points,I is the identity matrix,u is a column vector whose elements are all 1s, and ? denotes transposition.If the points lie in an m-dimensional spacethen C will be positive semide?ite, with rank m.Let F be any factoring of such a C, so that FF' = C.Then F contains the coordinates of the points.This is a well-known result in multidimensional scaling. === Is there a simple formula for the next Farey fraction?Let a/b be any irreducible rational number. Then, next Farey fraction isde?ed as the next element of the Farey sequence F_b. For example, nextFarey fraction of 3/5 is 2/3. Indeed:F_5 = {0/1,1/5,...,3/5,2/3,...,1/1}In other words, is there closed-form formula for elements of F_b? === > Is there a simple formula for the next Farey fraction?> Let a/b be any irreducible rational number. Then, next Farey fraction is> de?ed as the next element of the Farey sequence F_b. For example, next> Farey fraction of 3/5 is 2/3. Indeed:> F_5 = {0/1,1/5,...,3/5,2/3,...,1/1}> In other words, is there closed-form formula for elements of F_b?Here's a GP-Pari program which ?ds Farey sequences:{term(n,s,t)=local(z);z=?n+denominator(s))/ denominator(t));return((z*numerator(t)-numerator(s))/(z* denominator(t)-denominator(s)));}{farey(n)=local(f);f=[];if(n <2,return(f););if(n==2,f=concat(f,[1/2]);return(f););f=concat (f,[1/n]);f=concat(f,[1/(n-1)]);i=3;until(term(n,f[i-2],f[i-1 ])>=1,f=concat(f,[term(n,f[i-2],f[i-1])]);i=i+1);return(f);} farey(n) is the sequence with denominator n.F_5 = [1/5, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5]It was a little bit of time to come up with the right programming for thislittle gem. === On Thu, 15 Jan 2004 14:19:41 -0800, Mikito Harakiri>Is there a simple formula for the next Farey fraction?>Let a/b be any irreducible rational number. Then, next Farey fraction is>de?ed as the next element of the Farey sequence F_b. For example, next ^^^^^^^Huh?!?So you're assuming that the environment in which a/b lives is theFarey sequence of order b. If that is what you want to investigate,?e! But in the general case your problem is either not well posed orbadly phrased for you'd need, loosely speaking, *two* inputs: forexample (but not necessarily!) a rational number a/b and an integerorder, say n.Michele-- > Comments should say _why_ something is being done.Oh? My comments always say what _really_ should have happened. :)- Tore Aursand on comp.lang.perl.misc === > On Thu, 15 Jan 2004 14:19:41 -0800, Mikito Harakiri>Is there a simple formula for the next Farey fraction?>Let a/b be any irreducible rational number. Then, next Farey fraction is>de?ed as the next element of the Farey sequence F_b. For example, next> ^^^^^^^> Huh?!?> So you're assuming that the environment in which a/b lives is the> Farey sequence of order b. If that is what you want to investigate,> ?e! But in the general case your problem is either not well posed or> badly phrased for you'd need, loosely speaking, *two* inputs: for> example (but not necessarily!) a rational number a/b and an integer> order, say n.Michele,I don't understand your comment. Was I sloppy with the de?ition? Or thefact that the problem formulation is not general enough upset you? === > Is there a simple formula for the next Farey fraction?> Let a/b be any irreducible rational number. Then, next Farey fraction is> de?ed as the next element of the Farey sequence F_b. For example, next> Farey fraction of 3/5 is 2/3. Indeed:> F_5 = {0/1,1/5,...,3/5,2/3,...,1/1}> In other words, is there closed-form formula for elements of F_b?> In Knuth, _The Art of Computer Programming_, Section 1.3.2, ex. 19shows how to generate the next element of the Farey sequence from _two_successive elements. So you need two ?starters' to get going.It is a little harder to get the second ?starter' if you have only onefraction to begin with. But it is possible.De?ition: If two fractions a/b and c/d satisfy a/b < c/d, then Iwill say that they are *neighbors* if bc-ad=1. Proposition: Two fractions a/b and c/d are successive elements of theFarey series of order N if and only if these conditions are met:(a) a/b and c/d are neighbors;(b) b+d > N, b<=N, d<=N.Proposition: if a/b and c/d are neighbors, then a/b and (c+ka)/(d+kb)are neighbors.So, given a/b, we want to ?d c/d such that bc-ad=1. If our ?stsuccess at this fails to satisfy (b) in the ?st proposition, then wecan construct a more suitable neighbor by applying the secondproposition.Of course we put a/b in lowest terms, if it is not so already.Well, look at the equation to be solved in this form:ad = -1 + bc.This is as much as to say that -d is the multiplicative inverse of a inthe integers modulo b. since a is relatively prime to b, there is suchan inverse, an dit can be found by hacking around with euclid'salgorithm.This is not a closed form, but it is a terminating algorithm.-- Chris HenrichYes, one can rant about the program designs, but generally things keep gettingmore and more confused as time goes on. --Sea Wasp === >Is there a simple formula for the next Farey fraction?>Let a/b be any irreducible rational number. Then, next Farey fraction is>de?ed as the next element of the Farey sequence F_b. For example, next>Farey fraction of 3/5 is 2/3. Indeed:>F_5 = {0/1,1/5,...,3/5,2/3,...,1/1}>In other words, is there closed-form formula for elements of F_b?> In Knuth, _The Art of Computer Programming_, Section 1.3.2, ex. 19> shows how to generate the next element of the Farey sequence from _two_> successive elements. So you need two ?starters' to get going.also in Graham, Knuth, Patashnik (2nd ed), ch 4, Bonus problem 61, p. 150:(where N = b?)y0 = 0 y1 = 1 yt = y(t-1) ?t-N)/x(t-1)) - y(t-2)x0 = 1 x1 = N xt = x(t-1) ?t-N)/x(t-1)) - x(t-2)and the t'th Farey fraction based on N is y(t)/x(t). That this works is just an application of the usual gcd and mediant properties. > It is a little harder to get the second ?starter' if you have only one> fraction to begin with. But it is possible.or the dumb slow version, just scan through F_n.-- Mitch Harris(remove q to reply) === Is there a simple formula for the next Farey fraction?>Let a/b be any irreducible rational number. Then, next Farey fraction is>de?ed as the next element of the Farey sequence F_b. For example, next>Farey fraction of 3/5 is 2/3. Indeed:>F_5 = {0/1,1/5,...,3/5,2/3,...,1/1}>In other words, is there closed-form formula for elements of F_b?> In Knuth, _The Art of Computer Programming_, Section 1.3.2, ex. 19> shows how to generate the next element of the Farey sequence from _two_> successive elements. So you need two ?starters' to get going.> also in Graham, Knuth, Patashnik (2nd ed), ch 4, Bonus problem 61, p. 150:> (where N = b?)> y0 = 0 y1 = 1 yt = y(t-1) ?t-N)/x(t-1)) - y(t-2)> x0 = 1 x1 = N xt = x(t-1) ?t-N)/x(t-1)) - x(t-2)> and the t'th Farey fraction based on N is y(t)/x(t). That this works is> just an application of the usual gcd and mediant properties.> It is a little harder to get the second ?starter' if you have only one> fraction to begin with. But it is possible.> or the dumb slow version, just scan through F_n.My ?st attempt wasfunction next_farey ( l Rational ) RETURN Rational DETERMINISTIC IS exausted_fractions_search EXCEPTION;BEGIN FOR d In 1..l.den LOOP FOR c In 1..d LOOP if l.num*d+1=l.den*c then return new Rational(c,d); end if; END LOOP; END LOOP; raise exausted_fractions_search;END;but double loop is really killing: ?ding next_farey(17771,28345) took 300sec.Removing inner loop (at the expence of introduced division)function next_farey ( l Rational ) RETURN Rational DETERMINISTIC IS exausted_fractions_search EXCEPTION;BEGIN FOR d In 1..l.den LOOP if mod(l.num*d+1,l.den)=0 then return new Rational((l.num*d+1)/l.den,d); end if; END LOOP; raise exausted_fractions_search;END;is more reasonable implementation.I wonder if I can ?d next farey fraction faster than in linear time. === > Is there a simple formula for the next Farey fraction?> Let a/b be any irreducible rational number. Then, next Farey fraction is> de?ed as the next element of the Farey sequence F_b. For example, next> Farey fraction of 3/5 is 2/3. Indeed:> F_5 = {0/1,1/5,...,3/5,2/3,...,1/1}> In other words, is there closed-form formula for elements of F_b?Missed the obvious:min { c/d | ad+1=bc & a/b Then there is the idea that orthodox quantum theory with signal locality > is only a limited approximation like global Special Relativity and the > more general theory of the quantum information principle, including > quantum gravity, has signal nonlocality in a very essential way and > that all inner consciousness requires signal nonlocality in the sense > de?ed by Antony Valentini. Whoa! Did I just hear you suggest that orthodox quantum mechanicsjust might be bunk?:-)Bjacoby === james Randi owes me $1,000,000 . . . i think he is a cool guy and knowsWHAT IS UP!. read my threads on the STRAIGHTDOPE-MESSAGEBOARD, i am theposter who was Question. htttp://boards.straightdope.com....ignorance=educationgo to the SDMB and you can ?d some of the stupidest people on earth(notn insult, your ego creates suffering) . . the Internet is where ends willunite to form a whole. === Old rule, he who puts up the money makes the rules !> james Randi owes me $1,000,000 . . . i think he is a cool guy and knows> WHAT IS UP!. read my threads on the STRAIGHTDOPE-MESSAGEBOARD, i am the> poster who was Question.> htttp://boards.straightdope.com> .> .> .> .> ignorance=education> go to the SDMB and you can ?d some of the stupidest people on earth(not> n insult, your ego creates suffering) . . the Internet is where ends will> unite to form a whole.> === On a supercalifragilisticexpialidocious day, after dancing about singing Bibbety bobbety boo!, reotpreeoj ishkabibbled:^james Randi owes me $1,000,000 . . . i think he is a cool guy and knows^WHAT IS UP!. read my threads on the STRAIGHTDOPE-MESSAGEBOARD, i am the^poster who was Question. ^^htttp://boards.straightdope.com^^^The Straight Dope and its author, Cecil, totally ROCK! I really dig the kind of obscure info he digs up.-- The Queen of DXers, as well asQueen of the Commonwealth of Virginia, as well asThe Ruler of A.D.P., as well asSaint Debbe, as well asOur Lady of the Black Hole Exploratory Input Services as OhFishAlly Appointed by the Psychedelic Pope, a/k/a Saint Isidore of SevilleAn Ointed Minister of the Universal Life ChurchReverant of the Church of the SubGenius, UnOrthodoxSuperior Mutha Superior of the Little Sistahs of the Politically IncorrectWorshipper of Eris, Goddess of DiscordI WON'T grow up!! -- Peter === Craig Alan Feinstein: The Collatz 3n+1 Conjecture is Unprovable> I didn't know proving something about a function implies calculating> its value for every possible argument. You always learn something new.I didn't know that it was possible to show that a proof is wrong bylaughing at it without even finding any holes in it.Craig === > Algebraic toplogy (AT herein) seems to be a good way of formalizing> protocols in distributed systems (DS herein) (such as decision> problems like ?consensus'). I am interested in learning more, however> AT is a huge ?ld and I am only interested in learning the parts> directly related to distributed computing. Can anyone suggest a book,> that a) assumes no knowledge of algebraic toplogy b) assumes no more> than undergraduate level math - ie: calculus, linear algebra, basic> geometry, ability to do proofs, etc... c) focused on showing how AT> can be utilized to solve DS problems and does not get into non-DS> related aspects of AT (unless they are required background for> understanding DS related AT topics). > I have found many introductions on the net, but they seem to assume at> least basic knowledge of topology, homotopy, and other topics I am not> familiar with, so I think a full book dedicated to the subject sounds> more feasible as a basis for learning AT for DS. But any links or> online books you may know of will be of great help as well.> l8r, Mike N. Christoff> I thought I'd give an extremely low detail explanation of how algebraictopology can be to model asynchronous distributed protocols. I will also note that apparently it requires very little general topology to understand as it is almost entirely combinatorial in nature. In fact, one only needs to know the AT described in the ?st chapter of Munkres' book on algebraic topology to understand it (but not to understand this overview which requires none). This is taken from a response I gave on the theory-edge yahoo group (also posted on comp-sci-theory). Note that this must be viewed with a ?ed width font like courier or courier new.-- Original Message --major applications.--There has already been major work done in the connection between DCand AT. Many new lower bounds have been proven for classic DCproblems like k-set agreement and consensus. A quote by Fr.8ed.8ericTronel of the French National Institute for Reseasrch In ComputerScience and Control:... the problem of consensus (distributed agreement) plays a centralrole in the theory of distributed computing. Since it has been provento be unsolvable by deterministic algorithms in a purely asynchronoussystem, it was a real challenge to explicitly determine the borderthat exists between solvable and unsolvable problems in asynchronoussystems. This problem remained unsolved till the publication of aseminal paper by Maurice Herlihy and Nir Shavit. In fact at thereading of this paper, one can understand that all the previousapproaches where prone to fail. Indeed, all of them kept on using thetraditional way of modelling an asynchronous system, namely by themean of a graph of local states. This technic can be successful, whenthere is only one crash in the system. However, it does not scalewell, when the number of crashes increases. On the contrary, Herlihyand Shavit have chosen to model the evolution of the system throughthe use of high dimensional geometrical objects.The area of distributed computing I'm interested uses asynchronous message passing. This is the type you want to look at if you're interested in technologies like peer to peer networking (which I am).The idea is very combinatorial. You start with a set of nodes, eachwith an initial value from some ?ite set. For concreteness, imagine you have three nodes arranged in a triangle (vertices = nodes, edges = network connections). Let the initial value set = {0,1}. Then this can be represented asa -- b / / / cwhere a,b,c are tuples (node_id, initial_value). The number of possibleinitial states for the system are then 2^3 since we have 3 nodes which can each be initialized with either a 0 or a 1. This gives us 2^3 triangles.Now here is the tricky part. Connect the triangles together by matchingedges whose vertices a,b are equal. For example, take the three triangles below.A) 001(n1, 0) (n2,0) / / / (n3,1)B) 101(n1, 1) (n2,0) / / / (n3,1)C) 000(n1, 0) (n2,0) / / / (n3,0)This becomes: (n3,0) / / C / (n1, 0) (n2,0) / A / B / (n3,1) -- (n1,1)Now if we continue this process we will have generated a closedgeometric ?ure. This one in particular looks like like twotetrahedrons connected at their bases. Or one could think of it as anapproximation of a sphere using 8 triangles. This is called the inputcomplex at step 0.As the protocol evolves in time, each face (triangle) of the objectsubdivides further to represent all possible states the nodes could beat step 2. So for instance, the triangle C will subdivide into asubcomplex representing all possible states of the nodes afterstarting with the inital inputs all 0. The system is fault tolerant,as some of the structure of the subcomplex also represents possiblestates of the nodes if one or more of the nodes have failed.The set of structures that develop as the protocol progresses arecalled protocol complexes.In theoretical computer science we have the decsion problem, so in DCwe have decision tasks. A decision task is a ?ite process afterwhich each node makes a decision on what its output should be. Forexample, in the binary consensus problem each node is initially givena number in {0,1}.After a ?ite number of steps, all nodes must output identical valuescomplex of the problem. For binary consensus with three nodes, theonly acceptable output complex is shown below:(n1, 1) (n2,1) / / / (n3,1)(n1, 0) (n2,0) / / / (n3,0)In other words, the ?al state of all nodes must either be all 1s orall 0s.A ?simplicial map' is used to map the protocol complex to the outputcomplex. Without getting into the machinery of it (which I'm stillnot very clear on), the idea is to determine if the protocol complexcan evolve to the required output complex. This is done by trying to?d any topological obstructions (holes) to the simplicial map. Ifthere are, then the protocol cannot work, otherwise it can.That is a bare bare bones description of the idea and I may havemisexplained parts.l8r, Mike N. Christoff === I'm reposting this as a new thread since Outlook Express users in particular will probably never see it as a reply to the ?st thread since it will be buried almost a month back. Sorry to google users who see posts as they are posted in the main screen and not in threads initially.> Algebraic toplogy (AT herein) seems to be a good way of formalizing> protocols in distributed systems (DS herein) (such as decision> problems like ?consensus'). I am interested in learning more, however> AT is a huge ?ld and I am only interested in learning the parts> directly related to distributed computing. Can anyone suggest a book,> that a) assumes no knowledge of algebraic toplogy b) assumes no more> than undergraduate level math - ie: calculus, linear algebra, basic> geometry, ability to do proofs, etc... c) focused on showing how AT> can be utilized to solve DS problems and does not get into non-DS> related aspects of AT (unless they are required background for> understanding DS related AT topics). > I have found many introductions on the net, but they seem to assume at> least basic knowledge of topology, homotopy, and other topics I am not> familiar with, so I think a full book dedicated to the subject sounds> more feasible as a basis for learning AT for DS. But any links or> online books you may know of will be of great help as well.> l8r, Mike N. Christoff> I thought I'd give an extremely low detail explanation of how algebraictopology can be to model asynchronous distributed protocols. I will also note that apparently it requires very little general topology to understand as it is almost entirely combinatorial in nature. In fact, one only needs to know the AT described in the ?st chapter of Munkres' book on algebraic topology to understand it (but not to understand this overview which requires none). This is taken from a response I gave on the theory-edge yahoo group (also posted on comp-sci-theory). Note that this must be viewed with a ?ed width font like courier or courier new.-- Original Message --major applications.--There has already been major work done in the connection between DCand AT. Many new lower bounds have been proven for classic DCproblems like k-set agreement and consensus. A quote by Fr[Hyphen]d[Hyphen]ricTronel of the French National Institute for Reseasrch In ComputerScience and Control:... the problem of consensus (distributed agreement) plays a centralrole in the theory of distributed computing. Since it has been provento be unsolvable by deterministic algorithms in a purely asynchronous system, it was a real challenge to explicitly determine the borderthat exists between solvable and unsolvable problems in asynchronoussystems. This problem remained unsolved till the publication of aseminal paper by Maurice Herlihy and Nir Shavit. In fact at thereading of this paper, one can understand that all the previousapproaches where prone to fail. Indeed, all of them kept on using thetraditional way of modelling an asynchronous system, namely by themean of a graph of local states. This technic can be successful, whenthere is only one crash in the system. However, it does not scalewell, when the number of crashes increases. On the contrary, Herlihyand Shavit have chosen to model the evolution of the system throughthe use of high dimensional geometrical objects.The area of distributed computing I'm interested uses asynchronous message passing. This is the type you want to look at if you're interested in technologies like peer to peer networking (which I am).The idea is very combinatorial. You start with a set of nodes, eachwith an initial value from some ?ite set. For concreteness, imagine you have three nodes arranged in a triangle (vertices = nodes, edges = network connections). Let the initial value set = {0,1}. Then this can be represented asa -- b / / / cwhere a,b,c are tuples (node_id, initial_value). The number of possibleinitial states for the system are then 2^3 since we have 3 nodes which can each be initialized with either a 0 or a 1. This gives us 2^3 triangles.Now here is the tricky part. Connect the triangles together by matchingedges whose vertices a,b are equal. For example, take the three triangles below.A) 001(n1, 0) (n2,0) / / / (n3,1)B) 101(n1, 1) (n2,0) / / / (n3,1)C) 000(n1, 0) (n2,0) / / / (n3,0)This becomes: (n3,0) / / C / (n1, 0) (n2,0) / A / B / (n3,1) -- (n1,1)Now if we continue this process we will have generated a closedgeometric ?ure. This one in particular looks like like twotetrahedrons connected at their bases. Or one could think of it as anapproximation of a sphere using 8 triangles. This is called the inputcomplex at step 0.As the protocol evolves in time, each face (triangle) of the objectsubdivides further to represent all possible states the nodes could beat step 2. So for instance, the triangle C will subdivide into asubcomplex representing all possible states of the nodes afterstarting with the inital inputs all 0. The system is fault tolerant,as some of the structure of the subcomplex also represents possiblestates of the nodes if one or more of the nodes have failed.The set of structures that develop as the protocol progresses arecalled protocol complexes.In theoretical computer science we have the decsion problem, so in DCwe have decision tasks. A decision task is a ?ite process afterwhich each node makes a decision on what its output should be. Forexample, in the binary consensus problem each node is initially givena number in {0,1}.After a ?ite number of steps, all nodes must output identical valuescomplex of the problem. For binary consensus with three nodes, theonly acceptable output complex is shown below:(n1, 1) (n2,1) / / / (n3,1)(n1, 0) (n2,0) / / / (n3,0)In other words, the ?al state of all nodes must either be all 1s orall 0s.A ?simplicial map' is used to map the protocol complex to the outputcomplex. Without getting into the machinery of it (which I'm stillnot very clear on), the idea is to determine if the protocol complexcan evolve to the required output complex. This is done by trying to?d any topological obstructions (holes) to the simplicial map. Ifthere are, then the protocol cannot work, otherwise it can.That is a bare bare bones description of the idea and I may havemisexplained parts.l8r, Mike N. Christoff === > While I know C and C++, and have at times felt hampered by certain> limitations of Java, I've concluded that it's the best programming> language for wideranging discussions, as it's accessible to people who> haven't programmed before who aren't familiar with Unix, but are> Windows people.> For a nice C/C++ compiler/IDE package, you might want to check out http://www.bloodshed.net and download dev-C++.-- Will Twentyman === In sci.math, James Harrison 14 Jan 2004 08:34:22 -0800<3c65f87.0401140834.1d59810c@posting.google.com>:> While I know C and C++, and have at times felt hampered by certain> limitations of Java, I've concluded that it's the best programming> language for wideranging discussions, as it's accessible to people who> haven't programmed before who aren't familiar with Unix, but are> Windows people.> I need as many people capable of checking things like programs that I> post as I can get, so I'm making a push for Java.> It turns out that for most of you it's as simple as going to the Sun> website, where you can get a free download:> http://java.sun.com/j2se/1.4.2/download.html> I'd suggest Sun's tutorials for those who aren't at all familiar with> Java and running Java programs.> To see some of my work, and a program that you can try running, see my> blog archives:> You can also do a search at Google Groups for PrimeCountH.java,> The point for me here is to show you ?st that what I've discovered> works, and then I think it'll be easier for me to explain unique> features of my discovery, as I look towards ?ding more ef?ient and> effective ways of getting the word out.> James HarrisDid you *really* want me to run Yet Another Primecounter Contest? :-PAs it is, your Java implementation makes snails lookfast; on my Of?ial Test System(tm) it took 3 minutesto compute pi(10000)=1229 (which, to its credit, is thecorrect answer).You may want to look at your usage of Math.sqrt().A recoding of my ?st LegendrePhi submission took half a second,(to run, that is; the recoding itself took the better part ofa half hour) and of course also came up with the correct answer.And remember, I'm a lightweight in this department -- if ChristianBau were to really sink his teeth into this particular subproblem,he'd make both of us look ridiculous. :-) (Which is admittedly?e by me; I can't say I understand his algorithm but it's *fast*!)-- #191, ewill3@earthlink.netIt's still legal to go .sigless. === >In sci.math, James Harris>on 14 Jan 2004 08:34:22 -0800><3c65f87.0401140834.1d59810c@posting.google.com>:> While I know C and C++, and have at times felt hampered by certain> limitations of Java, I've concluded that it's the best programming> language for wideranging discussions, as it's accessible to people who> haven't programmed before who aren't familiar with Unix, but are> Windows people.> I need as many people capable of checking things like programs that I> post as I can get, so I'm making a push for Java. It turns out that for most of you it's as simple as going to the Sun> website, where you can get a free download:> http://java.sun.com/j2se/1.4.2/download.html> I'd suggest Sun's tutorials for those who aren't at all familiar with> Java and running Java programs.> To see some of my work, and a program that you can try running, see my> blog archives:> You can also do a search at Google Groups for PrimeCountH.java,> The point for me here is to show you ?st that what I've discovered> works, and then I think it'll be easier for me to explain unique> features of my discovery, as I look towards ?ding more ef?ient and> effective ways of getting the word out.> James Harris>Did you *really* want me to run Yet Another Primecounter Contest? :-P>As it is, your Java implementation makes snails look>fast; on my Of?ial Test System(tm) it took 3 minutes>to compute pi(10000)=1229 (which, to its credit, is the>correct answer).My gosh, it's _that_ slow? I guess I wasn't paying attentionduring the contest...Just for giggles I whipped up a _totally_ brainless PrimePiin Python just now - it's nothing but a sieve, and not aclever sieve, a totally idiotic sieve, not particularly wellwritten either, just top-of-the-brainless-head code:def Sieve(ran): for d in range(2, (len(ran))/2+1): for j in range(2*d, len(ran), d): ran[j]=0def Primes(n): ran = range(n+1) Sieve(ran) return ?ter(lambda x: x > 1, ran) def PrimePi(n): return len(Primes(n))This is _Python_, an extremely fabulous language, butquite slow, being interpreted instead of compiled. And(on and admittedly fast machine, 2000GH/512MB)it takes much less than a second to ?d Pi(10000);didn't actually time anything, but if I sayprint PrimePi(10000)the 1229 appears with no perceptible delay. Presumablyit would be pretty slow for larger values, for reasons beyondthe understanding of some of us, but it gives Pi(100000)in less than two seconds...Maybe there _is_ some signi?ance to James' work here.Like if we had a contest for the slowest possible Pi(n),maybe.>You may want to look at your usage of Math.sqrt().>A recoding of my ?st LegendrePhi submission took half a second,>(to run, that is; the recoding itself took the better part of>a half hour) and of course also came up with the correct answer.>And remember, I'm a lightweight in this department -- if Christian>Bau were to really sink his teeth into this particular subproblem,>he'd make both of us look ridiculous. :-) (Which is admittedly>?e by me; I can't say I understand his algorithm but it's *fast*!)David C. Ullrich === > This is _Python_, an extremely fabulous language, but> quite slow, being interpreted instead of compiled. I don't think you can prove programming skill by saying crazy thingslike Python is an extremely fabulous language. Sorry.c === On 15 Jan 2004 10:39:26 -0800, tb+usenet@becket.net (Thomas Bushnell, This is _Python_, an extremely fabulous language, but> quite slow, being interpreted instead of compiled. >I don't think you can prove programming skill by saying crazy things>like Python is an extremely fabulous language. Sorry.Well, I've already suggested that you should explain what yourreasons for this are. It occured to me that I instead of trying toexplain what's so great about Python I should just give a citationto someone else's explanation.I'm not an expert programmer, never claimed to be. On theother hand Eric Raymond is a _hugely_ respected name inopen-source programming circles. Go here to see what hehas to say about Python:A few quotes. First, just to show that he's not like the guyhe was very skeptical at ?st:>I immediately tripped over the ?st odd feature of Python >that everyone notices: the fact that whitespace (indentation) >is actually signi?ant in the language syntax. The language >has no analog of the C and Perl brace syntax; instead, >changes in indentation delimit statement groups. And, like >most hackers on ?st realizing this fact, I recoiled in >re? disgust. > I am just barely old enough to have programmed in >batch FORTRAN for a few months back in the 1970s. Most >hackers aren't these days, but somehow our culture seems >to have retained a pretty accurate folk memory of how nasty >those old-style ?ed-?ld languages were. Indeed, the >term ``free format'', used back then to describe the newer >style of token-oriented syntax in Pascal and C, has almost >been forgotten; all languages have been designed that way >for decades now. Or almost all, anyway. It's hard to blame >anyone, on seeing this Python feature, for initially reacting >as though they had unexpectedly stepped in a steaming pile >of dinosaur dung.He didn't start this project determined to like Python, his ?st reaction to one of the aspects that a lot of people ?d strangeat ?st was steaming pile of dinosaur dung.A lot of people who haven't _used_ the language react thatway. Now read what he has to say a little later, when hehas some experience with the language (and note howmuch experience I'm talking about):>Of course, this brought me face to face once again with >Python's pons asinorum, the signi?ance of whitespace. This >time, however, I charged ahead and roughed out some code >for a handful of sample GUI elements. Oddly enough, Python's >use of whitespace stopped feeling unnatural after about >twenty minutes. I just indented code, pretty much as I would >have done in a C program anyway, and it worked. > That was my ?st surprise. My second came a couple of >hours into the project, when I noticed (allowing for pauses >needed to look up new features in Programming Python) I was >generating working code nearly as fast as I could type. When >I realized this, I was quite startled. An important measure >of effort in coding is the frequency with which you write >something that doesn't actually match your mental>representation of the problem, and have to backtrack on >realizing that what you just typed won't actually tell the >language to do what you're thinking. An important measure >of good language design is how rapidly the percentage of >missteps of this kind falls as you gain experience with >the language. > When you're writing working code nearly as fast as >you can type and your misstep rate is near zero, it >generally means you've achieved mastery of the language. >But that didn't make sense, because it was still day >one and I was regularly pausing to look up new language >and library features! > This was my ?st clue that, in Python, I was >actually dealing with an exceptionally good design. Most >languages have so much friction and awkwardness built into >their design that you learn most of their feature set >long before your misstep rate drops anywhere near zero. >Python was the ?st general-purpose language I'd ever >used that reversed this process.That's a few _hours_ of experience using the language, and hisopinion has changed from steaming pile of dinosaur dung toan exceptionally good design.A few days later he's trying to do some metaclass hacking(there's an explanation of what that means in the original),and he determines that he can do things easily in Pythonthat he simply can't do at all in most languages, andcannot do easily in _any_ other language that he knows(he says he knows 20 or so):>This kind of thing is called metaclass hacking and is generally >considered fearsomely esoteric--deep black magic. Most >object-oriented languages don't support it at all; in >those that do (Perl being one), it tends to be a complicated >and fragile undertaking. I had been impressed by Python's >low coef?ient of friction so far, but here was a real test. >How hard would I have to wrestle with the language to get it >to do this?> I knew from previous experience that the bout was >likely to be painful, even assuming I won, but I dived into >the book and read up on Python's metaclass facilities. The >resulting function is shown in Listing 3, and the code that >calls it is in Listing 4. > it? Thirty-two lines, counting comments. Just from knowing >what I've said about the class structure, the calling code >is even readable. But the size of this code isn't the real >shocker. Brace yourself: this code only took me about ninety >minutes to write--and it worked correctly the ?st time >I ran it. > To say I was astonished would have been positively >wallowing in understatement. It's remarkable enough when >implementations of simple techniques work exactly as >expected the ?st time; but my ?st metaclass hack in >a new language, six days from a cold standing start? >Even if we stipulate that I am a fairly talented hacker, >this is an amazing testament to Python's clarity and >elegance of design.Could be that I'm not showing programming expertise by sayingcrazy things like Python is an extremely fabulous language. Ifso then _Eric Raymond_ is also revealing the same patheticlack of programming expertise. [A]mazing testament to Python'sclarity and elegance of design - what a moron.(Just for the record, there's some stuff towards the end that'sout of date: He explains he'd still use Perl for small thingsthat use regular expressions heavily. Evidently he was usingan early version of Python - the regular expressions incurrent Python are essentially identical to the regularexpressions in Perl.)>cDavid C. Ullrich === > Well, I've already suggested that you should explain what your> reasons for this are. It occured to me that I instead of trying to> explain what's so great about Python I should just give a citation> to someone else's explanation.Eric Raymond actually is not hugely respected by me. And heck, I'm ahugely respected name in open-source programming circles. Isn't proofby authority fun?More to the point: Python is not good for programming large systems.Most importantly here, Eric simply doesn't have experience in largeprogramming projects.Is Python better than C? I have no idea. C sucks too, though ofcourse for very different reasons. The things that Eric says arewonderful about Python are even more wonderful about Common Lisp orScheme, so I would advise using those languages instead, if you thinkthat Eric's argument is compelling.> Could be that I'm not showing programming expertise by saying> crazy things like Python is an extremely fabulous language. If> so then _Eric Raymond_ is also revealing the same pathetic> lack of programming expertise. [A]mazing testament to Python's> clarity and elegance of design - what a moron.Yes, Eric does not have much programming expertise. Sorry, but true. Thomas === > More to the point: Python is not good for programming large systems.Aside from the fact that this statement is complete nonsense, youhave a good point.More to the point, even if your statement were true it would notmean that Python was bad. There is a lot more to programming thanlarge systems . - William HughesBoy, we're *WAY* off topic for sci.math now. On the other hand this is a JSH thread.Sometimes ya just gotta say, What the heck. === On 16 Jan 2004 09:22:14 -0800, tb+usenet@becket.net (Thomas Bushnell, Well, I've already suggested that you should explain what your> reasons for this are. It occured to me that I instead of trying to> explain what's so great about Python I should just give a citation> to someone else's explanation.>Eric Raymond actually is not hugely respected by me. And heck, I'm a>hugely respected name in open-source programming circles. Isn't proof>by authority fun?>More to the point: Python is not good for programming large systems._Is_ that more to the point? This started when I posted 10 lines ofcode for the sake of doing one simple computation. _Supposing_ that in fact Python is not good for programming large systems, howdoes that show that my using it to solve one tiny problem showsa lack of expertise?Really. It may well be that Python is an extremely fabulous languagefor programming large systems would have been a crazy thing tosay. But that's not what I said - I said Python is an extremely fabulous programming language. There's a slight difference betweenthose two statements; the vast majority of programs that peoplewrite are not part of large systems.Note that the following question is a _question_. I'm not disputingyour statement that Python is not good for programming largesystems (I _am_ disputing the idea that that fact, if true, showsthat Python is an extremely fabulous language is a crazy thingto say.) Question, just for my information: why is Python not goodfor large systems?>Most importantly here, Eric simply doesn't have experience in large>programming projects.>Is Python better than C? I have no idea. The statements Python is better than C and C is better than Pythonboth strike me as ridiculous; asking whether Python is betterthan C is like asking whether a hammer is better than a screwdriver.For many purposes Python is much better than C. For many otherpurposes trying to use Python instead of C would be utterly stupid.(For example, trying to write a record-breaking PrimePi(n) calculatorin Python would probably be stupid (at least doing it in straightPython would be, having no experience with numpy I couldn'tsay how well suited it would be for something like that. If youthought that's what I was trying to do you didn't read my postvery completely, in particular you must have missed theJust for giggles I whipped up a _totally_ brainless PrimePiin Python just now - it's nothing but a sieve, and not aclever sieve, a totally idiotic sieve, not particularly wellwritten either, just top-of-the-brainless-head code: On theother hand, if for some reason I wanted to know the valueof PrimePi(10000) and didn't have a CAS handy thenspending a few minutes or hours writing a C program forthe purpose instead of a few seconds or minutes on aPython program seems equally stupid.))>C sucks too, though of>course for very different reasons. The things that Eric says are>wonderful about Python are even more wonderful about Common Lisp or>Scheme, so I would advise using those languages instead, if you think>that Eric's argument is compelling.I'm somewhat skeptical of what you say about Common Lisp andScheme - _all_ the things he says are wonderful about Pythonare even more wonderful about those languages? Do peopletalk about how they're writing working Common List or Scheme to solve actual problems they want to solve a few hours afterinstalling the language? Do those languages allow the samesort of transparent introspection with the same sort of ease?But never mind that, I don't know, so you may well be right aboutthose two languages. For the record, I wasn't convinced byEric's argument - long after I started using Python I sawconsistent with my own experience.(Long ago someone told me to use Perl for something. So Ilearned a little Perl - I hated it, the whole thing just made nosense. I heard about Python viewing a language war on aPerl group - I tried it, and it took me a few days to becomeas pro?ient at Python as I'd become in Perl in about a month.So learning enough Perl to do a tiny cgi thing took me amonth, so I'm stupid, ?e - but then if a moron like mecould do the same thing in Python in a few days it seemsto _me_ (as well as to a lot of other people) that that sayssomething about Python.And I loved it - it all made perfect sense, once you knowa few things about how it works you get it, and you caneasily learn the rest because you already got it. It happensfairly often that I have some problem to solve - some aresmall, some look big to me but I suspect they're stillvery small by comparison with the large projects youmention above. Using Python, once I've ?ured outexactly what I want to do it's simply never been anytrouble to just _do_ it. Which is why I call it anextremely fabulous language...)> Could be that I'm not showing programming expertise by saying> crazy things like Python is an extremely fabulous language. If> so then _Eric Raymond_ is also revealing the same pathetic> lack of programming expertise. [A]mazing testament to Python's> clarity and elegance of design - what a moron.>Yes, Eric does not have much programming expertise. Sorry, but true. Fine.>ThomasDavid C. Ullrich === > On 16 Jan 2004 09:22:14 -0800, tb+usenet@becket.net (Thomas Bushnell, > Note that the following question is a _question_. I'm not disputing> your statement that Python is not good for programming large> systems On the other hand I will dispute this statement.Python is good for programming large systems.>Is Python better than C? I have no idea. > The statements Python is better than C and C is better than Python> both strike me as ridiculous; asking whether Python is better> than C is like asking whether a hammer is better than a screwdriver.> For many purposes Python is much better than C. For many other> purposes trying to use Python instead of C would be utterly stupid.> Hey, I have an idea. Let's combine Python and C. Use Pythonfor the complicated high level stuff where speed of development is importantbut execution speed isn't, and C for the simpler low level stuff whereexecution speed is important. Wow, what a concept, combine an interpretedlanguage with a compiled language. I'm brilliant,I'm going to be famous, I'm going to be rich ....What do you mean someone else thought of this ?st?Oh well, send back the Rolls.> (For example, trying to write a record-breaking PrimePi(n) calculator> in Python would probably be stupid (at least doing it in straight> Python would be, having no experience with numpy I couldn't> say how well suited it would be for something like that.Numpy will allow you to do a lot of matrix stuff quickly withouthaving to go to C (or equivalent). However, if you have somethingspecialized that needs to be done quickly, Python/numpy is notthe way to go. - William Hughes <878yk9m4yp.fsf@becket.becket.net> === This is _Python_, an extremely fabulous language, but> quite slow, being interpreted instead of compiled. > I don't think you can prove programming skill by saying crazy things> like Python is an extremely fabulous language. Sorry.You're right. Real nerds denigrate every programming tool they ?d.The ultimate in geek cool is hating everything involving computers(while still playing with computers 23 hours a day, of course).-- What I've learned is that [mathematicians are] the gatekeepers, andseem to have almost absolute power when it comes to mathematics. -- James Harris, on All I Really Ever Needed to Know I Learned in /Ghostbusters/. === > You're right. Real nerds denigrate every programming tool they ?d.> The ultimate in geek cool is hating everything involving computers> (while still playing with computers 23 hours a day, of course).Huh? No, I simply mean that people don't right large systems inPython, and for very good reasons. <878yk9m4yp.fsf@becket.becket.net> <87wu7s8nil.fsf@phiwumbda.org> <878yk7yfoi.fsf@becket.becket.net> === You're right. Real nerds denigrate every programming tool they ?d.> The ultimate in geek cool is hating everything involving computers> (while still playing with computers 23 hours a day, of course).> Huh? No, I simply mean that people don't right large systems in> Python, and for very good reasons.>I didn't mean to imply that *you* had that attitude, but your commentreminded me of this tendency among geeks. Earlier today, I happenedto receive this fortune, which I think is very nice.One is not superior merely because one sees the world as odious. -- Chateaubriand (1768-1848)-- Many argue that its programmers have turned out shoddy programs, but[their] objective is to make pro?, not superlative programs perse. By the pro? criterion, Microsoft has been one of the greatestcompanies in the history of this country. -- ADTI defends Microsoft === > Huh? No, I simply mean that people don't right large systems in> Python, and for very good reasons.Do they left them?-- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.htmlNeedless to say, I had the last laugh. Alan Partridge, _Bouncing Back_ (14 times) === > Huh? No, I simply mean that people don't right large systems in> Python, and for very good reasons.> Do they left them?Oh blech, my ?gers slipped. :-PPerhaps they do upset them in Python, however... === > This is _Python_, an extremely fabulous language, but> quite slow, being interpreted instead of compiled. > I don't think you can prove programming skill by saying crazy things> like Python is an extremely fabulous language. Sorry.> Oh boy a language war. Can I play?Perl resembles line noise.C++ sucks moon rocks through a staw badly(but in a semi object oriented manner).Only an absolute moron would design a language(e.g. Python) where white space is signi?ant.The existence of the Obfuscated C Contest tellsyou all you need to know about C.LISP is favoured by the AI comunity. That'sbecause no one with real intelligence would ever use it.Tcl is an abomination unto the heavens.Pascal isn't even any good as a teaching language.COBOL?, ADA?, One acronym DOD!The teaching of BASIC rots the mind and assuch should be considered a criminal offence. (Dykstra)FORTRAN, bringing you 1950's technology today.APL .backwards is language ThisStick Java where the SUN doesn't shine.There is a persistent rumour that someone fed a SNOBOL compiler FORTRAN source anddidn't get a syntax error.MATLAB. Despite decades of effort, a Lapackwrapper is still a Lapack wrapper._______ It really doesn't matter what you put upon the list. They'd none of them be missed, They'd none of them be missedWATIAC, a decimal machine language, with op codesthat look suspiciously like octal, simulatedin COBOL. Now there's a language forreal progammers! - William Hughes === > This is _Python_, an extremely fabulous language, but> quite slow, being interpreted instead of compiled. > I don't think you can prove programming skill by saying crazy things> like Python is an extremely fabulous language. Sorry.>Oh boy a language war. Can I play?>Perl resembles line noise.>C++ sucks moon rocks through a staw badly>(but in a semi object oriented manner).>Only an absolute moron would design a language>(e.g. Python) where white space is signi?ant.Well of course your post was not meant to be takenseriously (or rather it seems you may have a seriouspoint, but these comments are not meant to betaken literally). But being a zealot I nonethelesscan't refrain from saying that people who actually_believe_ that the whitespace thing is bad shouldsee.>The existence of the Obfuscated C Contest tells>you all you need to know about C.>LISP is favoured by the AI comunity. That's>because no one with real intelligence would >ever use it.>Tcl is an abomination unto the heavens.>Pascal isn't even any good as a teaching language.>COBOL?, ADA?, One acronym DOD!>The teaching of BASIC rots the mind and as>such should be considered a criminal offence. (Dykstra)>FORTRAN, bringing you 1950's technology today.>APL .backwards is language This>Stick Java where the SUN doesn't shine.>There is a persistent rumour that someone >fed a SNOBOL compiler FORTRAN source and>didn't get a syntax error.>MATLAB. Despite decades of effort, a Lapack>wrapper is still a Lapack wrapper.>_______ It really doesn't matter what you put upon the list.> They'd none of them be missed,> They'd none of them be missed>WATIAC, a decimal machine language, with op codes>that look suspiciously like octal, simulated>in COBOL. Now there's a language for>real progammers!> - William HughesDavid C. Ullrich === >Only an absolute moron would design a language>(e.g. Python) where white space is signi?ant.> Well of course your post was not meant to be taken> seriously (or rather it seems you may have a serious> point, but these comments are not meant to be> taken literally).I didn't have a serious point, I just threw outa bunch of languages and insulted each one. Insome cases I used the canonical insult for the languague.For Python the canonical insult is the use of signi?antwhite space.I am a Python fan. I don't particularly like the signi?antwhite space, but it doesn't bother me either.The biggest problem is that the signi?ant white space issuedistracts from other much more important issues.(It's a bit like what would happen if the ANSI committeedecided to enforce the One True Brace Stylein standard C). - William HughesMaybe we should switch this to comp.lang.c People here justdon't get upset enough about off topic posts. === In sci.math, William Hugheson 15 Jan 2004 14:36:10 -0800<4d5e4663.0401151436.665cb562@posting.google.com>:> This is _Python_, an extremely fabulous language, but> quite slow, being interpreted instead of compiled. > I don't think you can prove programming skill by saying crazy things> like Python is an extremely fabulous language. Sorry.> Oh boy a language war. Can I play?> Perl resembles line noise.> C++ sucks moon rocks through a staw badly> (but in a semi object oriented manner).> Only an absolute moron would design a language> (e.g. Python) where white space is signi?ant.> The existence of the Obfuscated C Contest tells> you all you need to know about C.> LISP is favoured by the AI comunity. That's> because no one with real intelligence would > ever use it.> Tcl is an abomination unto the heavens.> Pascal isn't even any good as a teaching language.> COBOL?, ADA?, One acronym DOD!> The teaching of BASIC rots the mind and as> such should be considered a criminal offence. (Dykstra)> FORTRAN, bringing you 1950's technology today.> APL .backwards is language This> Stick Java where the SUN doesn't shine.> There is a persistent rumour that someone > fed a SNOBOL compiler FORTRAN source and> didn't get a syntax error.> MATLAB. Despite decades of effort, a Lapack> wrapper is still a Lapack wrapper.> _______ It really doesn't matter what you put upon the list.> They'd none of them be missed,> They'd none of them be missed WATIAC, a decimal machine language, with op codes> that look suspiciously like octal, simulated> in COBOL. Now there's a language for> real progammers!> - William HughesHey, you forgot C#, Visual Basic, ASP, most versions of assembly,Modula-2, Modula-3, Ruby, SQL, WATFOR and WATFIV (though they'rereally dialects of Fortran), Postscript (yes, it's a programminglanguage!) and FORTH... :-)C#: Tomorrow's technology today. Maybe. Wait until the bugpatch, which will be out sometime, erm, next month. Yeah, next month. Oh, and it's sort of like Java except not really. We're not sure yet.Visual Basic: It's pretty basic, and it only can be used visually.ASP: It bites. Hard.Assembly: Sometimes it's required. Most times it should sit there in little pieces on the ?as that's what it looks like.Modula-2: Pascal squared.Modula-3: Pascal cubed.Ruby: Is it a gem of a language? I don't know, really.SQL: Please, someone, write one.WATFOR: ... and give it what for. At least FORTRAN has an excuse.WATFIV: A SQL to WATFOR. And you know what sequels do...Postscript/FORTH: Both of these are backwards, too, although I'll admit there's a certain elegance to 1 5 + . as it doesn't need parentheses. Of course LISP took them all anyway...TeX: An interesting entry, actually, and I think it does qualify as a computer language although that's a bit like saying an automobile quali?s as a guerney. (Or vice versa.)JCL: You don't want to know. Fortunately, Unix usurped ?dd' and made it a lot clearer -- which for Unix is amazing... :-)DCL: VMS's answer to shell scripts, complete with F$PARSE().Bourne: Unix's answer to how do I do this as cryptically as possible?REXX: IBM's answer to shell scripts, and probably someone's dog's name. There are times it looks like the end product thereof.HTML: LISP with angle brackets.XML: LISP with angle brackets and question marks.XHTML: The worst of both.MathML: Like XHTML only mathematical.XSL: De?itely in excess.ASN.1: Otherwise known as Abstruse, Strange, and Non-comprehensible. Even parsing the MIB can give one a headache. The sad thing is: this might very well be the most ef?ient, absent compression.ASN.2: I hope not.SOAP: Was it ever simple? How is XML an object? How does one access an object using XML? This one should be put out of its misery.-- #191, ewill3@earthlink.netIt's still legal to go .sigless. === > Hey, you forgot C#, Visual Basic, ASP, most versions of assembly,> Modula-2, Modula-3, Ruby, SQL, WATFOR and WATFIV (though they're> really dialects of Fortran), Postscript (yes, it's a programming> language!)Look, if you are going to include things that weren't intended to beused as a general purpose programming language but can be so usedwe have to include most text editors, most word processors, mostspreadsheets, etc. etc. etc. we'll never ?ish!. Of coursewe'll never ?ish even if we stick to general purpose programminglanguages but this is even worse. It's sort of like the differencebetween the in?ity of the integers and the in?ity of the reals(note the valiant attempt to get back on topic). - William HughesP.S. Have I explained the self evident error in logic made by the evil cantorians? and FORTH... > :-)> C#: Tomorrow's technology today. Maybe. Wait until the bugpatch,> which will be out sometime, erm, next month. Yeah, next month.> Oh, and it's sort of like Java except not really. We're not> sure yet.> Visual Basic: It's pretty basic, and it only can be used visually.> ASP: It bites. Hard.> Assembly: Sometimes it's required. Most times it should sit there> in little pieces on the ?as that's what it looks like.> Modula-2: Pascal squared.> Modula-3: Pascal cubed.> Ruby: Is it a gem of a language? I don't know, really.> SQL: Please, someone, write one.> WATFOR: ... and give it what for. At least FORTRAN has an excuse.> WATFIV: A SQL to WATFOR. And you know what sequels do...> Postscript/FORTH: Both of these are backwards, too, although I'll admit> there's a certain elegance to> 1 5 + .> as it doesn't need parentheses. Of course LISP took them> all anyway...> TeX: An interesting entry, actually, and I think it does qualify as> a computer language although that's a bit like saying an automobile> quali?s as a guerney. (Or vice versa.)> JCL: You don't want to know. Fortunately, Unix usurped ?dd'> and made it a lot clearer -- which for Unix is amazing... :-)> DCL: VMS's answer to shell scripts, complete with F$PARSE().> Bourne: Unix's answer to how do I do this as cryptically as possible?> REXX: IBM's answer to shell scripts, and probably someone's dog's name.> There are times it looks like the end product thereof.> HTML: LISP with angle brackets.> XML: LISP with angle brackets and question marks.> XHTML: The worst of both.> MathML: Like XHTML only mathematical.> XSL: De?itely in excess.> ASN.1: Otherwise known as Abstruse, Strange, and Non-comprehensible.> Even parsing the MIB can give one a headache. The sad thing> is: this might very well be the most ef?ient, absent> compression.> ASN.2: I hope not.> SOAP: Was it ever simple? How is XML an object? How does one> access an object using XML? This one should be put out of its misery. === > JCL: You don't want to know. Fortunately, Unix usurped ?dd'> and made it a lot clearer -- which for Unix is amazing... :-)I was exposed to JCL once. My therapist says I am makinggood progress. - William Hughes === HTML: LISP with angle brackets.> XML: LISP with angle brackets and question marks.> XHTML: The worst of both.these aren't even programming languages (ie. Turing complete).!Q === On 15 Jan 2004 10:39:26 -0800, tb+usenet@becket.net (Thomas Bushnell, This is _Python_, an extremely fabulous language, but> quite slow, being interpreted instead of compiled. >I don't think you can prove programming skill by saying crazy things>like Python is an extremely fabulous language. Sorry.Well, actually if you look back you'll see I wasn't trying to proveI was a skilled programmer, my point was almost the opposite,that it takes essentially no skill to beat the benchmark that hadbeen mentioned._But_ surely you're not serious. Python is in fact an extremelyfabulous language. Not for every purpose, but for _many_purposes it's the best thing out there.Before I start rambling about what's so great about Python,why don't you explain what's not so great about it? Mightbe more ef?ient if I knew what to refute. (Some complaintsmy reply would be no, it's not a fabulous language for thatpurpose, other complaints I'd insist it _was_ fabulous andtry to explain why. What complaint did you have in mind?)>cDavid C. Ullrich === >In sci.math, James Harris>on 14 Jan 2004 08:34:22 -0800><3c65f87.0401140834.1d59810c@posting.google.com>:> While I know C and C++, and have at times felt hampered by certain> limitations of Java, I've concluded that it's the best programming> language for wideranging discussions, as it's accessible to people who> haven't programmed before who aren't familiar with Unix, but are> Windows people.> I need as many people capable of checking things like programs that I> post as I can get, so I'm making a push for Java. It turns out that for most of you it's as simple as going to the Sun> website, where you can get a free download:> http://java.sun.com/j2se/1.4.2/download.html> I'd suggest Sun's tutorials for those who aren't at all familiar with> Java and running Java programs.> To see some of my work, and a program that you can try running, see my> blog archives:> You can also do a search at Google Groups for PrimeCountH.java,> The point for me here is to show you ?st that what I've discovered> works, and then I think it'll be easier for me to explain unique> features of my discovery, as I look towards ?ding more ef?ient and> effective ways of getting the word out.> James Harris>Did you *really* want me to run Yet Another Primecounter Contest? :-P>As it is, your Java implementation makes snails look>fast; on my Of?ial Test System(tm) it took 3 minutes>to compute pi(10000)=1229 (which, to its credit, is the>correct answer).> My gosh, it's _that_ slow? I guess I wasn't paying attention> during the contest...> Just for giggles I whipped up a _totally_ brainless PrimePi> in Python just now - it's nothing but a sieve, and not a> clever sieve, a totally idiotic sieve, not particularly well> written either, just top-of-the-brainless-head code:> def Sieve(ran):> for d in range(2, (len(ran))/2+1):> for j in range(2*d, len(ran), d):> ran[j]=0> def Primes(n):> ran = range(n+1)> Sieve(ran)> return ?ter(lambda x: x > 1, ran)> def PrimePi(n):> return len(Primes(n))> This is _Python_, an extremely fabulous language, but> quite slow, being interpreted instead of compiled. And> (on and admittedly fast machine, 2000GH/512MB)> it takes much less than a second to ?d Pi(10000);> didn't actually time anything, but if I say print PrimePi(10000)> the 1229 appears with no perceptible delay. Presumably> it would be pretty slow for larger values, for reasons beyond> the understanding of some of us, but it gives Pi(100000)> in less than two seconds... Maybe there _is_ some signi?ance to James' work here.Well, if I understand James' argument, _your_ program doesn't give youa partial differential equation, so _you_ won't be able to solve the ReimannHypothesis. ;)> Like if we had a contest for the slowest possible Pi(n),> maybe.>You may want to look at your usage of Math.sqrt().>A recoding of my ?st LegendrePhi submission took half a second,>(to run, that is; the recoding itself took the better part of>a half hour) and of course also came up with the correct answer.>And remember, I'm a lightweight in this department -- if Christian>Bau were to really sink his teeth into this particular subproblem,>he'd make both of us look ridiculous. :-) (Which is admittedly>?e by me; I can't say I understand his algorithm but it's *fast*!)> David C. Ullrich === > Well, if I understand James' argument, _your_ program doesn't give you> a partial differential equation, so _you_ won't be able to solve the Reimann> Hypothesis. ;)I'm still waiting for James to show any sign that he has the foggiestidea what the Riemann Hypothesis *is*...-- Wayne Brown (HPCC #1104) | When your tail's in a crack, you improvisefwbrown@bellsouth.net | if you're good enough. Otherwise you give | your pelt to the trapper.e^(i*pi) = -1 -- Euler | -- John Myers Myers, Silverlock === In sci.math, David C Ullrichon Thu, 15 Jan 2004 05:25:26 -0600:In sci.math, James Harris><3c65f87.0401140834.1d59810c@posting.google.com>:> While I know C and C++, and have at times felt hampered by certain> limitations of Java, I've concluded that it's the best programming> language for wideranging discussions, as it's accessible to people who> haven't programmed before who aren't familiar with Unix, but are> Windows people. I need as many people capable of checking things like programs that I> post as I can get, so I'm making a push for Java.> It turns out that for most of you it's as simple as going to the Sun> website, where you can get a free download:> http://java.sun.com/j2se/1.4.2/download.html> I'd suggest Sun's tutorials for those who aren't at all familiar with> Java and running Java programs.> To see some of my work, and a program that you can try running, see my> blog archives:> You can also do a search at Google Groups for PrimeCountH.java,> The point for me here is to show you ?st that what I've discovered> works, and then I think it'll be easier for me to explain unique> features of my discovery, as I look towards ?ding more ef?ient and> effective ways of getting the word out.> James Harris>Did you *really* want me to run Yet Another Primecounter Contest? :-P>As it is, your Java implementation makes snails look>fast; on my Of?ial Test System(tm) it took 3 minutes>to compute pi(10000)=1229 (which, to its credit, is the>correct answer).> My gosh, it's _that_ slow? I guess I wasn't paying attention> during the contest...I was referring to his Java implementation, that he's postedhttp://home.earthlink.net/~ewill3/math/ primecounters/:-)It doesn't match either of his two submissions for my contestso I'm a little puzzled as to what he's thinking. To itscredit it has no memoization or special cases. (Of course,that may be its problem...)> Just for giggles I whipped up a _totally_ brainless PrimePi> in Python just now - it's nothing but a sieve, and not a> clever sieve, a totally idiotic sieve, not particularly well> written either, just top-of-the-brainless-head code:> def Sieve(ran):> for d in range(2, (len(ran))/2+1):> for j in range(2*d, len(ran), d):> ran[j]=0> def Primes(n):> ran = range(n+1)> Sieve(ran)> return ?ter(lambda x: x > 1, ran)> def PrimePi(n):> return len(Primes(n))> This is _Python_, an extremely fabulous language, but> quite slow, being interpreted instead of compiled. And> (on and admittedly fast machine, 2000GH/512MB)> it takes much less than a second to ?d Pi(10000);> didn't actually time anything, but if I say print PrimePi(10000)> the 1229 appears with no perceptible delay. Presumably> it would be pretty slow for larger values, for reasons beyond> the understanding of some of us, but it gives Pi(100000)> in less than two seconds...My system ran it in .311 seconds.> Maybe there _is_ some signi?ance to James' work here.> Like if we had a contest for the slowest possible Pi(n),> maybe.Naw. Christian Bau got that, erm, honor. Of course, that'smostly because he was coding for extreme simplicity in thisalgorithm, and wasn't trying to be the fastest. (It's a verysimple, elegant, and rather stupid algorithm; basically, itloops over the integers from 1 to N then a subloop tests forprimality in the simplest way possible using the modulo (?%')operator.)But when computing pi(10000), it's faster than my sieve...and a *lot*faster than James' Java. (0.051 seconds)A recoding into Java resulted in a .61s run.don't know how well it'll work for pi(10000).http://www.cbau.freeserve.co.uk/for the details thereon.>You may want to look at your usage of Math.sqrt().>A recoding of my ?st LegendrePhi submission took half a second,>(to run, that is; the recoding itself took the better part of>a half hour) and of course also came up with the correct answer.>And remember, I'm a lightweight in this department -- if Christian>Bau were to really sink his teeth into this particular subproblem,>he'd make both of us look ridiculous. :-) (Which is admittedly>?e by me; I can't say I understand his algorithm but it's *fast*!) David C. Ullrich-- #191, ewill3@earthlink.netIt's still legal to go .sigless. === Been too long since Ive looked at this stuff.If I have two coordinates (300, 1500) and (500, 4000),Given any x, what is the *#$@ formula to calculate the corresponding y (orx, given y) between these two points? === In sci.math, BCC<42GNb.11269$3q7.9922@newssvr25. news.prodigy.com>:> Been too long since Ive looked at this stuff.> If I have two coordinates (300, 1500) and (500, 4000),> Given any x, what is the *#$@ formula to calculate the corresponding y (or> x, given y) between these two points?> There are a few ways of solving this; the simplest is arguablycreating a parameter t and linear functions x(t) and y(t)such that:x(0) = 300x(1) = 500y(0) = 1500y(1) = 4000It turns out x(t) = 300 + (500-300)*t and y(t) = 1500+(4000-1500)*t,or x(t) = 300+200*t and y(t) = 1500+2500*t.One can then substitute:x(t) = 300+200*t implies t(x) = (x-300)/200y = 1500 + 2500*(x-300)/200 = 1500 + 12.5*x - 3750 = 12.5*x - 2250As a check, feeding x = 300 gives us y = 1500and x = 500 gives us y = 4000, as it should.Another method is to equate y = ax+b and solve for a and b;one gets two equations in two unknowns by plugging in thetwo above coordinate points.-- #191, ewill3@earthlink.netIt's still legal to go .sigless. === > Been too long since Ive looked at this stuff.> If I have two coordinates (300, 1500) and (500, 4000),> Given any x, what is the *#$@ formula to calculate the corresponding y (or> x, given y) between these two points?> This is called the two-point form of the equation of a line.If the line passes through (x1, y1) and (x2, y2), thenall points on the line (x, y) satisfy(y-y1)/(x-x1) = (y2-y1)/(x2-x1).If x1=x2, then the points are (x1, y) for any y.Similarly if y1=y2.To get a feel for what is going on, plot a few cases. === BCC> Been too long since Ive looked at this stuff.> If I have two coordinates (300, 1500) and (500, 4000),> Given any x, what is the *#$@ formula to calculate the corresponding y (or> x, given y) between these two points?If the equation isy = ax + bthen we need1500 = a*300 + b4000 = a*500 + bfrom which we get a = 12.5 and b = -2250. So, the equation isy = 12.5 x - 2250.LH === I have one number, zero, that I like better than others.0's the solution to every algebraic equation. That's why 0's my favorite number.What's your favorite?And why?Garry Denke, GeologistDenoco Inc. of Texas === > I have one number, zero, that I like better than others.> 0's the solution to every algebraic equation. > That's why 0's my favorite number.> What's your favorite?> And why?> Garry Denke, Geologist> Denoco Inc. of Texaspeanut butter and jelly.Charlie Volkstorf === > I have one number, zero, that I like better than others. > 0's the solution to every algebraic equation. > That's why 0's my favorite number. > What's your favorite? > And why? > peanut butter and jelly.Thinking about it, the closest number I get is 57, but I do not evenknow whether that is valid.-- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === > I have one number, zero, that I like better than others.> 0's the solution to every algebraic equation.> That's why 0's my favorite number.> What's your favorite?> And why?> peanut butter and jelly.> Thinking about it, the closest number I get is 57, but I do not even> know whether that is valid.26 = number of required dimensions for string theory to work.l8r, Mike N. Christoff === > I have one number, zero, that I like better than others.> 0's the solution to every algebraic equation. That's why 0's my favorite > number.> What's your favorite?My favorite is 17.> And why?It's the ?st random number.Frank === > I have one number, zero, that I like better than others.> 0's the solution to every algebraic equation. That's why 0's my favorite > number.> What's your favorite?> My favorite is 17.> The third Fermat number, or the ?st to end in 7.> And why?> It's the ?st random number.> FrankWhat makes it random?-- === > I have one number, zero, that I like better than others.> 0's the solution to every algebraic equation. That's why 0's my favorite > number.> What's your favorite?> My favorite is 17.> And why?> It's the ?st random number.> FrankI was going to say 137, since it proves the existence of God and givesmeaning to the universe... but I like Frank's answer better.Rickwork@ostrander.de === > I have one number, zero, that I like better than others.> 0's the solution to every algebraic equation. That's why 0's myfavorite> number.> What's your favorite?> My favorite is 17.> And why?> It's the ?st random number.> Frank> I was going to say 137, since it proves the existence of God and gives> meaning to the universeI thought that number was 42? Damn! My entire system of beliefs isshattered! Oh well. All praise the mighty 137!!l8r, Mike N. Christoff === > I have one number, zero, that I like better than others.> 0's the solution to every algebraic equation. That's why 0's my favorite > number.> What's your favorite?> My favorite is 17.> And why?> It's the ?st random number.> FrankDoes anybody really believe Gregory Chaitin that Mathematics israndom? How could computers work, then? I think he's prettyconfused.1. Non-deterministic (random) = life2. Deterministic = science3. Deterministic and doesn't involve your 5 senses (input) =Mathematics4. Deterministic but not calculable = Theory of ComputationChaitin is confusing # 1 and # 4.BTW: In my Axiomatic Basis for Everything (all of Logic, Mathematicsand Computer Science) the fundamental axiom is I#yes(I) that meansthat it's all deterministic (not random.) The # means solves orcreates when evaluated and yes(I) means the set of numbers thatprogram (Turing Machine) number I halts on, so that program I createsthe set of inputs on which it halts yes.That's actually the axiom for the Theory of Computation. For English(e.g. the Liar Paradox) it's I#ts(I) which says that English sentenceI evaluates to the truth of I.In general we have I#run(I) but in each case it is saying thatevaluating an object is deterministic.Charlie Volkstorf === > 1. Non-deterministic (random) = lifeNot quite. Life is not random. It is merely complex.> 2. Deterministic = scienceNot quite. Science is not deterministic. It merely predicts with suf?ient accuracy for suf?ient number of cases.> 3. Deterministic and doesn't involve your 5 senses (input) => MathematicsNot quite. Mathematics = assumptions => cognition => senses.> 4. Deterministic but not calculable = Theory of ComputationNot Quite. Theory of computation is nothing more than the concept of addition.> Charlie VolkstorfMany Happy Returns of Janus. === > 1. Non-deterministic (random) = life> Not quite. Life is not random. It is merely complex.Is there anything that is random?> 2. Deterministic = science> Not quite. Science is not deterministic. It merely predicts > with suf?ient accuracy for suf?ient number of cases.If something is not deterministic, then how can science relate to it?> 3. Deterministic and doesn't involve your 5 senses (input) => Mathematics> Not quite. Mathematics = assumptions => cognition => senses.Which senses would you need to develop or perform mathematics?> 4. Deterministic but not calculable = Theory of Computation> Not Quite. Theory of computation is nothing more than the > concept of addition.Then why don't they call it Arithmetic?> Charlie Volkstorf> Many Happy Returns of Janus. === > 1. Non-deterministic (random) = life> Not quite. Life is not random. It is merely complex.>I'm not saying I agree/disagree but this is still an open question.> 2. Deterministic = science> Not quite. Science is not deterministic. It merely predicts> with suf?ient accuracy for suf?ient number of cases.>Ok.> 3. Deterministic and doesn't involve your 5 senses (input) => Mathematics> Not quite. Mathematics = assumptions => cognition => senses.>I think what Charlie was getting at is that math (as it stands today) is notempirical. However the idea of ?empirical math' / ?experimental math' isstarting to come together. Though largely by today's standards, the veryphrase ?empirical math' is an oxymoron.> 4. Deterministic but not calculable = Theory of Computation> Not Quite. Theory of computation is nothing more than the> concept of addition.>This is way off.l8r, Mike N. Christoff === > I have one number, zero, that I like better than others.> 0's the solution to every algebraic equation. That's why 0's my favorite > number.> What's your favorite?0/0It's just so playful!Charlie VolkstorfSo what all are the properties of 0 (and 1, for that matter) and doesit have to be the same number for all of those properties? E.g. X+0=Xand X*0=0 but could there be two different numbers that do that? What's it about + and * that makes it the same number? === In sci.logic, Garry Denkeon 15 Jan 2004 16:26:13 -0800<4e63857.0401151626.7573b27b@posting.google.com>:> I have one number, zero, that I like better than others.> 0's the solution to every algebraic equation. > That's why 0's my favorite number.> What's your favorite?> And why?x^2 = 1 has a solution x = 0?Somehow, I doubt it.My favorite numbers are e, i, pi, and -1.They just interrelate so well together... :-) Garry Denke, Geologist> Denoco Inc. of Texas-- #191, ewill3@earthlink.netIt's still legal to go .sigless. === > In sci.logic, Garry Denke> on 15 Jan 2004 16:26:13 -0800> <4e63857.0401151626.7573b27b@posting.google.com>:> I have one number, zero, that I like better than others.> 0's the solution to every algebraic equation. > That's why 0's my favorite number.> What's your favorite?> And why?> x^2 = 1 has a solution x = 0?the knowledge of nothing is everything, sonothing squared plus one equals nothing.here, i will write the algebraic equation out so i can solve it for you. x^2 + 1 = 0solution for the equation is 0 the number(the number to the right of equals sign).challenge: post any algebraic equation in the form of it equaling 0 the number, and i will solve your algebraic equation.> Somehow, I doubt it.here are some more algebraic equations whose solution is 0 the number(the number to the right of the equals sign):radical equation... sqrt(x - 10) - 4 = 0quadratic equation... x^2 - 5x + 3 = 0exponential equation... e^2x - 3e^x + 2 = 0logarithmic equation... 6log(x^2 + 1) - x = 0trigonometric equation... 3tan^2 x - 1 = 0 > My favorite numbers are e, i, pi, and -1.> They just interrelate so well together...if you post it i will answer.garry denke, geologistdenoco inc. of texas === > In sci.logic, Garry Denke> on 15 Jan 2004 16:26:13 -0800> <4e63857.0401151626.7573b27b@posting.google.com>:> I have one number, zero, that I like better than others.> 0's the solution to every algebraic equation.> That's why 0's my favorite number.> What's your favorite?> And why?> x^2 = 1 has a solution x = 0?> the knowledge of nothing is everything, so> nothing squared plus one equals nothing.> here, i will write the algebraic equation out so i can solve it for you.> x^2 + 1 = 0> solution for the equation is 0 the number> (the number to the right of equals sign).> challenge: post any algebraic equation in the form of it equaling> 0 the number, and i will solve your algebraic equation.> Somehow, I doubt it.> here are some more algebraic equations whose solution is 0 the number> (the number to the right of the equals sign):Actually, 1 is the solution to every equation:radical equation... sqrt(x - 10) - 3 = 1quadratic equation... x^2 - 5x + 4 = 1exponential equation... e^2x - 3e^x + 3 = 1logarithmic equation... 6log(x^2 + 1) - x + 1 = 1trigonometric equation... 3tan^2 x = 1l8r, Mike N. Christoff === > I have one number, zero, that I like better than others.> 0's the solution to every algebraic equation. > That's why 0's my favorite number.> What's your favorite?> And why?> Garry Denke, Geologist> Denoco Inc. of TexasThe reason garry Dense loves zero so much is that it tells you everything there is to know about him. === --ZHANG Yanhttp://www.ntu.edu.sg/home5/pg01308021>Many thanks for the reply.>The pdf of Y can be determined independently from Z but depend on the pdfof>X. For example, the sample value y of Y is uniformly distributed in [0,x]>where x is the sample value of X.>The pdf of Z can be determined independently from Y but depend on the pdfof>X. For example, the sample value z of Z is uniformly distributed in [0,10 *>x] where x is the sample value of X.>That is, both of the sample values of Y and Z are reliant on the sample>value of X. Then,>1. are Y and Z dependent?>2. if pdf of Y and Z are given, can we precisely obtain the pdf of A?>3. any approximation algorithm to obtain the pdf of A from the pdf of Yand>Z by considering that Y and Z are virtually independent??> As I said before, though Y and Z are *conditionally* independent> gilven X, they are *not* in general independent (not even> virtually). In previous posts, I have given formulae which determine> the density of A. Here is another:> P{A>a} = EP(Y>a, Z>a | X) = E[P(Y>a | X) P(Z>a | X)]> = int(x=0..infty, int(w=a..infty, f(w|x)) int(w=a..infty, g(w|x)) h(x))> where f(.|x) and g(.|x) are respectively the conditional densities> of Y and Z given X=x and h is the marginal density of X.> (Again, I assume that X, Y, and Z are jointly continuous, and that> X is nonnegative.) Taking the derivative, the density of A at a is> int(x=0..infty, h(x) [f(a|x) int(w=a..infty, g(w|x)) + g(a|x)> int(w=a..infty, f(w|x))])> --> Stephen J. Herschkorn herschko@rutcor.rutgers.edu>X-Cise: tanbanso@iinet.net.auX-CompuServe-Customer: YesX-Coriate: admin@interspeed.co.nzX-Ecrate: tanandtanlawyers.comX-Pose: george_cox@btinternet.comX-Punge: Micro$oft === In , on 01/14/2004 at 09:37 AM, azilla said:> I know that the vector space R^n over R(real) is not>isomorphic to the vector space R over R. A book told me that if we>regard R^n and R as vector spaces over Q(rational), R^n is>isomorphic to R. Is that true?In what set theory? I assume that your book is using a set theory thatincludes the Axiom of Choice or an equivalent.>How does the book come up with that?Zorn's Lemma. You order all partial bases by inclusion, and take theunion of any maximal chain.-- Shmuel (Seymour J.) Metz, SysProg and JOATnot reply to spamtrap@library.lspace.org === > Dear all,> I know that the vector space R^n over R(real) is not isomorphic to> the vector space R over R. A book told me that if we regard R^n and R as> vector spaces over Q(rational), R^n is isomorphic to R. Is that true? How> But R^n and R are isomorphic as Q-vector spaces only way out there inAxiom of Choice Land. Without AC one cannot prove it. One cannot*actually write down* an isomorphism between R^2 and R.-- G. A. Edgar http://www.math.ohio-state.edu/~edgar/ === >Dear all,> I know that the vector space R^n over R(real) is not isomorphic to>the vector space R over R. A book told me that if we regard R^n and R as>vector spaces over Q(rational), R^n is isomorphic to R. Is that true? HowWith the Axiom of Choice, there is a basis for R over Q,and similarly this gives a basis for R^n over Q. As bothbases have the cardinality of the continuum, they areisomorphic.-- This address is for information only. I do not claim that these viewsare those of the Statistics Department or of Purdue University.Herman Rubin, Department of Statistics, Purdue UniversityX-Cise: tanbanso@iinet.net.auX-CompuServe-Customer: YesX-Coriate: admin@interspeed.co.nzX-Ecrate: tanandtanlawyers.comX-Pose: george_cox@btinternet.comX-Punge: Micro$oft === In , on 01/14/2004 at 09:09 AM, whopkins@csd.uwm.edu (Alfred Einstead) said:>In general, the algebra consisting of all C^{in?ity} functions>(i.e. functions continuously differentiable to all orders) for a>compact Hausdorff spaceHow do you de?e differentiable if you do not require that thespace be a manifold?-- Shmuel (Seymour J.) Metz, SysProg and JOATnot reply to spamtrap@library.lspace.org === > Can you brilliant and gifted Mathematicians and Physicists help out> this poor soul and justify your acceptance of relativistic and quantum> mechanics by providing the axioms of a simultaneously discrete and> continuous space.Look up the paper in the following link. It posits a stationary and elasticcontinuous space with light as discrete wave packets in this continuousmedium moving at constant c wrt to it.http://www.journaloftheoretics.com/Links/Papers/ Seto.pdfKen Seto === > Can you brilliant and gifted Mathematicians and Physicists help out> this poor soul and justify your acceptance of relativistic and quantum> mechanics by providing the axioms of a simultaneously discrete and> continuous space.> To go that route, you should shift focus away from point sets to> function algebras. For instance, a circle is equivalently> described by the following algebra generated from the elements:> ...,z_{-3}, z_{-2}, z_{-1}, z_0, z_1, z_2, z_3,...> with the multiplication rule:> z_a z_b = z_{a+b}> the corresponding functions being z_a <-> Z_a(x) = exp(2 pi i a x).> In general, the algebra consisting of all C^{in?ity} functions> (i.e. functions continuously differentiable to all orders) for> a compact Hausdorff space comprises a commutative C^* algebra> -- and vice versa (all commutative C^* algebras are function> spaces for some compact Hausdorff space).> To generalize to something that encompasses both continuous and> discrete spaces, you generalize the space of functions from> the C^{in?ity) functions to a suitable class of generalized> functions -- in particular, a class large enough to include> the delta functions.> Such a class, likewise, is to be closed under products (and> sums and multiplication by constants, as usual), closed> under differentiation, and is to include both the delta> functions and the C^{in?ity} functions.Does this math properly express the physical issue ?I know that quantities, which are related to the continuum, can havediscrete _values_ . I mean concretely integration constants frompartial differential equations. I could demonstrate that theintegration constants for most stable solutions of the Einstein-Thus, one must not provide the axioms of a simultaneously discrete andcontinuous space. Continuous time & space are enough to understandthat all.BTW, our friend Bill Hobba points out the Kochem-Specker theorem ateach occasion. If I understood it properly, it would even supportmy insights. May be, you have an opinion about it ?Ulrich === > In , on 01/12/2004> at 07:49 PM, j.schoenfeld@programmer.net (John Schoenfeld) said:>Can you brilliant and gifted Mathematicians and Physicists help out>this poor soul and justify your acceptance of relativistic and>quantum mechanics by providing the axioms of a simultaneously>discrete and continuous space.> First, the issue is one of Physics and not Mathematics. Second,> physicists don't normally do axioms. Third, there is nothing in either> SR or QT that would require, or even allow, a simultaneously discrete> and continuous space. QFT requires a spacetime that is a real> manifold. The perceived discreteness is in certain measured> quantities, not in spacetime.Agree fully. The certain measured quantities are exactly the?st integration constants of the Einstein-Maxwell equations.I could indeed demonstrate that the integration constants forhttp://home.t-online.de/home/Ulrich.Bruchholz/Ulrichinfo at bruchholz minus acoustics dot de === > Dear John Schoenfeld:> Can you brilliant and gifted Mathematicians and Physicists help out> this poor soul and justify your acceptance of relativistic and quantum> mechanics by providing the axioms of a simultaneously discrete and> continuous space.> If the only tool you have is a hammer, then all problems look like nails. Well that's the problem. When the only tool you have is a derivative, everything sounds like an integral.> But if you have two tools, say a hammer and a saw...> one that joins problem sets into a neat whole, and> one that cuts the problem into its contituent parts,> then you can apply the tool that is required. That does works. But since it only works in Dualistic Philosophy, it only works on the weekends.> The Universe is both discrete and continuous, depending on what you> need/want to describe. But I am not a brilliant and gifted Mathematician and Physicist. That's easy to explain though. The only gifted Mathematician who ever lived was semi-continous, rather than continous.> David A. Smith === > Dear John Schoenfeld:> Can you brilliant and gifted Mathematicians and Physicists help out> this poor soul and justify your acceptance of relativistic and quantum> mechanics by providing the axioms of a simultaneously discrete and> continuous space.> If the only tool you have is a hammer, then all problems look like nails.> Well that's the problem. When the only tool > you have is a derivative, everything sounds> like an integral.> But if you have two tools, say a hammer and a saw...> one that joins problem sets into a neat whole, and> one that cuts the problem into its contituent parts,> then you can apply the tool that is required.> That does works. But since it only works in > Dualistic Philosophy, it only works on the weekends.> The Universe is both discrete and continuous, depending on what you> need/want to describe.> But I am not a brilliant and gifted Mathematician and Physicist.> That's easy to explain though.> The only gifted Mathematician who ever lived > was semi-continous, rather than continous. Or as modern science is usually taught. The Physicist asked the Mathematician: Quote-Unquote: Is there a doctor in the house? May Spock be with you. The Mathematician answered: Slash-Unslash: No, but if you get the brain surgeon a male nurse, he'll fake it. May Bach be with you. > David A. Smith === > Can you brilliant and gifted Mathematicians and Physicists help out> this poor soul and justify your acceptance of relativistic and quantum> mechanics by providing the axioms of a simultaneously discrete and> continuous space.> Sure!> Awwww... ain't that cute? The two little cranks are playing together!You appear to be undergoing a massive emotional ?S === > It must be pointed out that Varney can't tell the difference between> deceleration and -acceleration (and he has a PhD).> There is no essential difference. Deceleration is just acceleration > multiplied by -1. Either one means a change in velocity wrt time.Deceleration means velocity magnitude is decreasing with time, whichis satis?d by the condition v.a < 0. Putting a negative symbolinfront of a does not imply decreasing velocity.> Since acceleration is a vector all that counts is magnitude and > direction. A vector is a vector is a vector. JS === Dear John Schoenfeld:...> There is no essential difference. Deceleration is just acceleration> multiplied by -1. Either one means a change in velocity wrt time.> Deceleration means velocity magnitude is decreasing with time, which> is satis?d by the condition v.a < 0. Putting a negative symbol> infront of a does not imply decreasing velocity.Not strictly true either, John. The actual de?tion should be more alongthe lines of:v x a = 0 and v . a < 0shouldn't it?David A. Smith === > Dear John Schoenfeld:> ...> There is no essential difference. Deceleration is just acceleration> multiplied by -1. Either one means a change in velocity wrt time.> Deceleration means velocity magnitude is decreasing with time, which> is satis?d by the condition v.a < 0. Putting a negative symbol> infront of a does not imply decreasing velocity.> Not strictly true either, John. The actual de?tion should be more along> the lines of:> v x a = 0 and v . a < 0> shouldn't it?I don't believe so, since the cross product does not work for alldimensionsn > 0 which is a requirement (as far as I know, that is)To decrease the velocity of a body, it must accelerate in a directiongreater than 90degrees to the direction of velocity. Although itshould be noted that a decelerating body always tends to a state ofacceleration such that a body can never be in?itely decelerating. > David A. SmithJS === > Dear John Schoenfeld:> ...> There is no essential difference. Deceleration is just acceleration> multiplied by -1. Either one means a change in velocity wrt time.> Deceleration means velocity magnitude is decreasing with time, which> is satis?d by the condition v.a < 0. Putting a negative symbol> infront of a does not imply decreasing velocity. Not strictly true either, John. The actual de?tion should be more along> the lines of:> v x a = 0 and v . a < 0> shouldn't it?>I don't believe so, since the cross product does not work for all>dimensions>n > 0 which is a requirement (as far as I know, that is)How many dimensions are there on your world? Cross-products can be generalized if they need to be.But I think v.a<0 itself is enough. Actually, .a, where is a unit vector in the direction of the velocity, =v/|v|, gives the magnitude of acceleration that changes speed, as opposed to an acceleration that changes direction at constant speed. If it's less than zero you're slowing down and can call it deceleration. >To decrease the velocity of a body, it must accelerate in a direction>greater than 90degrees to the direction of velocity.Yes, that's what .a < 0 says.-- Don't try to teach a pig how to sing. You'll waste your time and annoy the pig. === > Not strictly true either, John. The actual de?tion should be more along> the lines of:> v x a = 0 and v . a < 0> shouldn't it?>I don't believe so, since the cross product does not work for all>dimensions>n > 0 which is a requirement (as far as I know, that is)> How many dimensions are there on your world? Cross-products can be > generalized if they need to be.> But I think v.a<0 itself is enough. Correct.Proof:v.a < 0|v||a|cos(x) < 0cos(x) < 0x > cos^-1(0)x > pi/2 x > 90 degrees> Actually, .a, where is a unit > vector in the direction of the velocity, =v/|v|, gives the magnitude of > acceleration that changes speed,Correct because you taking the component of acceleration parallel tothe direction of velocity.> as opposed to an acceleration that changes direction at constant speed.Incorrect. a.v < 0 implies decreasing |v|, but does not say anything about a.v = 0 implies constant |v|, but changesa.v > 0 implies increasing |v|, but does not say anything about If it's less than zero you're slowing down and can call it deceleration.Correct. >To decrease the velocity of a body, it must accelerate in a direction>greater than 90degrees to the direction of velocity.> Yes, that's what .a < 0 says.You don't need , you just need v (see above proof).JS === Dear John Schoenfeld:> Not strictly true either, John. The actual de?tion should be morealong> the lines of:> v x a = 0 and v . a < 0> shouldn't it?>I don't believe so, since the cross product does not work for all>dimensions>n > 0 which is a requirement (as far as I know, that is)> How many dimensions are there on your world? Cross-products can be> generalized if they need to be.> But I think v.a<0 itself is enough.> Correct.> Proof:> v.a < 0> |v||a|cos(x) < 0> cos(x) < 0> x > cos^-1(0)> x > pi/2> x > 90 degreesWhich is no proof, since any value 180A >= x > 90A meets this requirement.And no net change in the magnitude of velocity need result for other valuesfor the angle. So v . a < 0 can be met and d|v|/dt is unconstrained.David A. Smith === > Dear John Schoenfeld:> Not strictly true either, John. The actual de?tion should be more> along> the lines of:> v x a = 0 and v . a < 0> shouldn't it?>I don't believe so, since the cross product does not work for all>dimensions>n > 0 which is a requirement (as far as I know, that is)> How many dimensions are there on your world? Cross-products can be> generalized if they need to be.> But I think v.a<0 itself is enough.> Correct.> Proof:> v.a < 0> |v||a|cos(x) < 0> cos(x) < 0> x > cos^-1(0)> x > pi/2> x > 90 degrees> Which is no proof, since any value 180A >= x > 90A meets this requirement.> And no net change in the magnitude of velocity need result for other values> for the angle. So v . a < 0 can be met and d|v|/dt is unconstrained.> David A. SmithMy last part of the proof was erroneous, let me correct that.Proof:v.a < 0|v||a|cos(x) < 0cos(x) < 0x > cos^-1(0) AND x < pi - cos^-1(0)pi/2 < x < 3pi/2So to decrease the velocity of an object, an acceleration must beapplied at between 90 and 270 degrees (exclusive) to the velocityvector.Trust me, I am right. The dot product of a vector function and it'sderivative that is less than 0 denotes a decreasing vector function.This proof is very long now (for me), but i will ?ish it eventually.JS === Dear John Schoenfeld:> Dear John Schoenfeld: Not strictly true either, John. The actual de?tion should bemore> along> the lines of:> v x a = 0 and v . a < 0> shouldn't it?>I don't believe so, since the cross product does not work for all>dimensions>n > 0 which is a requirement (as far as I know, that is)> How many dimensions are there on your world? Cross-products can be> generalized if they need to be.> But I think v.a<0 itself is enough.> Correct.> Proof:> v.a < 0> |v||a|cos(x) < 0> cos(x) < 0> x > cos^-1(0)> x > pi/2> x > 90 degrees> Which is no proof, since any value 180A >= x > 90A meets thisrequirement.> And no net change in the magnitude of velocity need result for othervalues> for the angle. So v . a < 0 can be met and d|v|/dt is unconstrained.> David A. Smith> My last part of the proof was erroneous, let me correct that.> Proof:> v.a < 0> |v||a|cos(x) < 0> cos(x) < 0> x > cos^-1(0) AND x < pi - cos^-1(0)> pi/2 < x < 3pi/2> So to decrease the velocity of an object, an acceleration must be> applied at between 90 and 270 degrees (exclusive) to the velocity> vector.> Trust me, I am right. The dot product of a vector function and it's> derivative that is less than 0 denotes a decreasing vector function.> This proof is very long now (for me), but i will ?ish it eventually.You are not right. d|v|/dt < 0 is *not* the same as v . a < 0There are too many degrees of freedom in n-dimensional space (with n > 1,and neglecting time as a dimension included in n). v x a = 0 *is* too draconian, requiring only colinear motion/action. ButI cannot ?ure out how to express deceleration *exactly* as anything otherthan d|v|/dt < 0. Perhaps you can put your massive intellect to the task.David A. Smith === Dear John Schoenfeld:> Dear John Schoenfeld:>in...> My last part of the proof was erroneous, let me correct that.> Proof:> v.a < 0> |v||a|cos(x) < 0> cos(x) < 0> x > cos^-1(0) AND x < pi - cos^-1(0)> pi/2 < x < 3pi/2> So to decrease the velocity of an object, an acceleration must be> applied at between 90 and 270 degrees (exclusive) to the velocity> vector.> Trust me, I am right. The dot product of a vector function and it's> derivative that is less than 0 denotes a decreasing vector function.> This proof is very long now (for me), but i will ?ish it eventually.> You are not right. d|v|/dt < 0 is *not* the same as v . a < 0Correction. You *are* right only when the velocity and acceleration arecolinear.> There are too many degrees of freedom in n-dimensional space (with n > 1,> and neglecting time as a dimension included in n).> v x a = 0 *is* too draconian, requiring only colinear motion/action.But> I cannot ?ure out how to express deceleration *exactly* as anythingother> than d|v|/dt < 0. Perhaps you can put your massive intellect to thetask.David A. Smith === > There are too many degrees of freedom in n-dimensional space (with n > 1,> and neglecting time as a dimension included in n).> v x a = 0 *is* too draconian, requiring only colinear motion/action.> But> I cannot ?ure out how to express deceleration *exactly* as anything> other> than d|v|/dt < 0. Perhaps you can put your massive intellect to the> task.> David A. SmithHere is the proof that v.a < 0 = deceleration.let v = velocity vectorlet a = acceleration vectorWe know that decleration occurs when d|v|/dt is a negative vector.If vector a0 is the projection of a onto v, then a0 = d|v|/dt.a0 = [(a.v) / |v|] a0 = d|v|/dtd|v|/dt = (a.v / |v|) = ( |a||v|cos(x) / |v| ) = |a|cos(x) Since a0 and v are colinear, we can reduce the problem to 1 dimension,and thus we can substitute with 1, as we shall let be thepositive axis of the dimension.So,d|v|/dt = |a|cos(x)We know that we are decelerating when d|v|/dt < 0, So,|a| cos(x) < 0cos(x) < 0pi/2 < x < 3pi/2So we have proven that the angle between the velocity and accelerationvectors must be interior to 90 degrees and 270 degrees fordeceleration to occur.Now we shall prove that (v.a) < 0 implies deceleration.|a|cos(x) = d|v|/dt |v||a|cos(x) = d|v|/dt |v|a.v = d|v|/dt |v|(a.v)/|v| = d|v|/dt(a.v)/|v| < 0(|a||v|cos(x))/|v| < 0cos(x) < 0pi/2 < x < 3pi/2Alas the same result.Interesting properties that you may not have noted:d|v|/dt = (a.v) / |v|, a 1-dimensional vectorJS === > Trust me, I am right. The dot product of a vector function and it'sits> derivative that is less than 0 denotes a decreasing vector function.> This proof is very long now (for me), but i will ?ish it eventually.It's very short for the rest of us. If v(t) has v(t).v'(t) < 0then (d/dt)[|v(t)|^2] = (d/dt)[v(t).v(t)] = 2 v(t).v'(t) < 0so |v(t)|^2 is decreasing and |v(t)| is decreasing.-- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.htmlNeedless to say, I had the last laugh. Alan Partridge, _Bouncing Back_ (14 times) === Dear Gregory L. Hansen:...> Not strictly true either, John. The actual de?tion should be morealong> the lines of:> v x a = 0 and v . a < 0> shouldn't it?>I don't believe so, since the cross product does not work for all>dimensions>n > 0 which is a requirement (as far as I know, that is)> How many dimensions are there on your world? Cross-products can be> generalized if they need to be.> But I think v.a<0 itself is enough. Actually, .a, where is a unit> vector in the direction of the velocity, =v/|v|, gives the magnitudeof> acceleration that changes speed, as opposed to an acceleration that> changes direction at constant speed. If it's less than zero you're> slowing down and can call it deceleration.If the velocity is in the x-direction, and the acceleration is in both thex and y direction (whatever the sign), the x component could decelerate,but the y would not. v . a < 1 would be satis?d, but velocity would nothave to be decreasing with time. No?So v . a < 1 is not suf?ient as a de?ition for deceleration. Thecross product is *required* to assure that no component of acceleration isin any other axis (in an n-dimensional Universe).David A. Smith === >Dear Gregory L. Hansen:>...> Not strictly true either, John. The actual de?tion should be more>along> the lines of:> v x a = 0 and v . a < 0> shouldn't it?>I don't believe so, since the cross product does not work for all>dimensions>n > 0 which is a requirement (as far as I know, that is)How many dimensions are there on your world? Cross-products can be> generalized if they need to be.But I think v.a<0 itself is enough. Actually, .a, where is a unit> vector in the direction of the velocity, =v/|v|, gives the magnitude>of> acceleration that changes speed, as opposed to an acceleration that> changes direction at constant speed. If it's less than zero you're> slowing down and can call it deceleration.>If the velocity is in the x-direction, and the acceleration is in both the>x and y direction (whatever the sign), the x component could decelerate,>but the y would not. v . a < 1 would be satis?d, but velocity would not>have to be decreasing with time. No?>So v . a < 1 is not suf?ient as a de?ition for deceleration. The>cross product is *required* to assure that no component of acceleration is>in any other axis (in an n-dimensional Universe).I assume you meant v . a < 0?If the velocity in in the x-direction and acceleration is in both the x accelerating even if it's also changing direction. For instance, if you're driving in a circle and hit the brakes. You don't have to go in a straight line to change your kinetic energy.a would not, in general, be the magnitude of deceleration, unless a x v = 0. Maybe that's what you meant, that a would be the actual amount of deceleration. But for general acceleration, a.v/|v| would be the magnitude of deceleration no matter what the transverse part is. That's just picking out the component that's parallel to the velocity, throwing away the transverse part. You could also ?d the tranverse acceleration, a x v/|v|, if you like.-- A nice adaptation of conditions will make almost any hypothesis agreewith the phenomena. This will please the imagination but does not advanceour knowledge. -- J. Black, 1803. === Dear Gregory L. Hansen:...> But I think v.a<0 itself is enough. Actually, .a, where is aunit> vector in the direction of the velocity, =v/|v|, gives themagnitude>of> acceleration that changes speed, as opposed to an acceleration that> changes direction at constant speed. If it's less than zero you're> slowing down and can call it deceleration.>If the velocity is in the x-direction, and the acceleration is in boththe>x and y direction (whatever the sign), the x component could decelerate,>but the y would not. v . a < 1 would be satis?d, but velocity wouldnot>have to be decreasing with time. No?>So v . a < 1 is not suf?ient as a de?ition for deceleration. The>cross product is *required* to assure that no component of accelerationis>in any other axis (in an n-dimensional Universe).> I assume you meant v . a < 0?Yes. Sorry.> If the velocity in in the x-direction and acceleration is in both the xLet's put some imaginary numbers to this. v = 100i, a = -1i + 14.1jAt this instant, v . a < 0, but at the end of 1 unit of time, |v| has thesame magnitude as this instant. So it couldn't be said to be decelerating.> That's decelerating or> accelerating even if it's also changing direction. For instance, if> you're driving in a circle and hit the brakes. You don't have to go in a> straight line to change your kinetic energy.I agree, but John's de?ition does not handle multi-dimensional problems.> a would not, in general, be the magnitude of deceleration, unless> a x v = 0. Maybe that's what you meant, that a would be the actualamount> of deceleration. But for general acceleration, a.v/|v| would be the> magnitude of deceleration no matter what the transverse part is. That's> just picking out the component that's parallel to the velocity, throwing> away the transverse part. You could also ?d the tranverse> acceleration, a x v/|v|, if you like.I'm just trying to ?d an absolute de?ition of deceleration. Incommon parlance that means the magnitude of the velocity is gettingsmaller. Regardless of coordinate frame.David A. Smith === >Dear Gregory L. Hansen:>...> But I think v.a<0 itself is enough. Actually, .a, where is a>unit> vector in the direction of the velocity, =v/|v|, gives the>magnitude>of> acceleration that changes speed, as opposed to an acceleration that> changes direction at constant speed. If it's less than zero you're> slowing down and can call it deceleration.>If the velocity is in the x-direction, and the acceleration is in both>the>x and y direction (whatever the sign), the x component could decelerate,>but the y would not. v . a < 1 would be satis?d, but velocity would>not>have to be decreasing with time. No?>So v . a < 1 is not suf?ient as a de?ition for deceleration. The>cross product is *required* to assure that no component of acceleration>is>in any other axis (in an n-dimensional Universe).I assume you meant v . a < 0?>Yes. Sorry. If the velocity in in the x-direction and acceleration is in both the x>Let's put some imaginary numbers to this. v = 100i, a = -1i + 14.1j>At this instant, v . a < 0, but at the end of 1 unit of time, |v| has the>same magnitude as this instant. So it couldn't be said to be decelerating.Well, sure, but the velocity vector will change, so a.v/|v| will change. It will be negative for some small time, and then go positive. But if a car were braking while turning a corner, the acceleration vector would also change versus time, it wouldn't be constant. That's decelerating or> accelerating even if it's also changing direction. For instance, if> you're driving in a circle and hit the brakes. You don't have to go in a> straight line to change your kinetic energy.>I agree, but John's de?ition does not handle multi-dimensional problems.Oh, well, I guess one dimension is pretty easy to handle then. Don't even need a dot product for that. a would not, in general, be the magnitude of deceleration, unless> a x v = 0. Maybe that's what you meant, that a would be the actual>amount> of deceleration. But for general acceleration, a.v/|v| would be the> magnitude of deceleration no matter what the transverse part is. That's> just picking out the component that's parallel to the velocity, throwing> away the transverse part. You could also ?d the tranverse> acceleration, a x v/|v|, if you like.>I'm just trying to ?d an absolute de?ition of deceleration. In>common parlance that means the magnitude of the velocity is getting>smaller. Regardless of coordinate frame.Good luck. I'll pull a few more numbers out of my butt. Assume an initial velocity of 10i and acceleration -1i. Velocity will be (10-1*t)i, slowing down for ten seconds and then speeding up. Boost to a frame x'=x+10*t. Initial velocity v'=0, there will be no period of deceleration.-- Is that plutonium on your gums?Shut up and kiss me! -- Marge and Homer Simpson === > Dear Gregory L. Hansen:> ...> But I think v.a<0 itself is enough. Actually, .a, where is a> unit> vector in the direction of the velocity, =v/|v|, gives the> magnitude> of> acceleration that changes speed, as opposed to an acceleration that> changes direction at constant speed. If it's less than zero you're> slowing down and can call it deceleration.>If the velocity is in the x-direction, and the acceleration is in both> the>x and y direction (whatever the sign), the x component could decelerate,>but the y would not. v . a < 1 would be satis?d, but velocity would> not>have to be decreasing with time. No?>So v . a < 1 is not suf?ient as a de?ition for deceleration. The>cross product is *required* to assure that no component of acceleration> is>in any other axis (in an n-dimensional Universe).> I assume you meant v . a < 0?> Yes. Sorry.> If the velocity in in the x-direction and acceleration is in both the x> Let's put some imaginary numbers to this. v = 100i, a = -1i + 14.1j> At this instant, v . a < 0, but at the end of 1 unit of time, |v| has the> same magnitude as this instant. So it couldn't be said to be decelerating.> That's decelerating or> accelerating even if it's also changing direction. For instance, if> you're driving in a circle and hit the brakes. You don't have to go in a> straight line to change your kinetic energy.> I agree, but John's de?ition does not handle multi-dimensional problems.> a would not, in general, be the magnitude of deceleration, unless> a x v = 0. Maybe that's what you meant, that a would be the actual> amount> of deceleration. But for general acceleration, a.v/|v| would be the> magnitude of deceleration no matter what the transverse part is. That's> just picking out the component that's parallel to the velocity, throwing> away the transverse part. You could also ?d the tranverse> acceleration, a x v/|v|, if you like.> I'm just trying to ?d an absolute de?ition of deceleration. In> common parlance that means the magnitude of the velocity is getting> smaller. Regardless of coordinate frame.A body is declerating when v.a < 0. Didn't they ever teach youSCHOENFELDS VECTOR MAGNITUDE THEOREM:The magnitude of the vector given by the vector function F(x) isdecreasing if F(x) . F^1(x) < 0, increasing if F(x) . F^1(x) > 0,constant if F(x) . F^1(x) = 0.Proof:Is much longer than I expected, so I'll get back to you on this one.JS === > Awwww... ain't that cute? The two little cranks are playing together!When I was working at MIT, a professor of my acquaintance (I'll sayThe idea was to be kind, and yet at the same time, not waste his owntime which was better spent on other things. He thought that justignoring people or dismissing them would be unkind. And since heworked at the AI lab and was high pro?e in various ways, he getslook at it, see what the topic was, and pull the ?e of other lettersabout the same topic. Then he would reply with a form letter sayingsomething like: I'm sorry, but I don't have the expertise to addressyour letter. The following people may be better suited to discussyour points. Then he would add the names and addresses of a few ofNow this struck me as wonderfully brilliant. Simultaneously beingkind, moreover, giving the nut someone to talk to who will listen andrespond in the same vein, satisfying both, keeping his own time freefor other things--amazing.Thomas === > Awwww... ain't that cute? The two little cranks are playing together!> When I was working at MIT, a professor of my acquaintance (I'll say> The idea was to be kind, and yet at the same time, not waste his own> time which was better spent on other things. He thought that just> ignoring people or dismissing them would be unkind. And since he> worked at the AI lab and was high pro?e in various ways, he gets> look at it, see what the topic was, and pull the ?e of other letters> about the same topic. Then he would reply with a form letter saying> something like: I'm sorry, but I don't have the expertise to address> your letter. The following people may be better suited to discuss> your points. Then he would add the names and addresses of a few of> Now this struck me as wonderfully brilliant. Simultaneously being> kind, moreover, giving the nut someone to talk to who will listen and> respond in the same vein, satisfying both, keeping his own time free> for other things--amazing.Have you ever noticed that cranks and nutters are harder toget together than Pandas during mating season? If you watchthem (the cranks) frolic on usenet, you'll notice that theyseem to, for the most part, utterly ignore each other. It'sas though they instinctively know to avoid their own species. === > Awwww... ain't that cute? The two little cranks are playing together!> When I was working at MIT, a professor of my acquaintance (I'll say> The idea was to be kind, and yet at the same time, not waste his own> time which was better spent on other things. He thought that just> ignoring people or dismissing them would be unkind. And since he> worked at the AI lab and was high pro?e in various ways, he gets> look at it, see what the topic was, and pull the ?e of other letters> about the same topic. Then he would reply with a form letter saying> something like: I'm sorry, but I don't have the expertise to address> your letter. The following people may be better suited to discuss> your points. Then he would add the names and addresses of a few of> Now this struck me as wonderfully brilliant. Simultaneously being> kind, moreover, giving the nut someone to talk to who will listen and> respond in the same vein, satisfying both, keeping his own time free> for other things--amazing.> Have you ever noticed that cranks and nutters are harder to> get together than Pandas during mating season? If you watch> them (the cranks) frolic on usenet, you'll notice that they> seem to, for the most part, utterly ignore each other. It's> as though they instinctively know to avoid their own species.That is a very astute observation. At most they will utilize thematerial that is posted by their fellow cranks for their ?needs. Thus none of them have the balls to criticize anything theothers post. If they did then they risk of getting into an argument.If they get into an argument then each knows that they will never beable to admit to making an error nor simply letting the other personsimply disagree or have another opinion and let it rest. Instead theyknow that if they get into a ?ar then they can't use thatperson's ?to ?thers. For example: varney will neverdiscuss physics to any deep extent. If he did and they disagreed thenone would, without doubt, start insulting the other and claim theirstupid etc. If that happens then varney will never be able to sayeveryone thinks your stupid because he's too scared of someonereminding him that one of the people he is referring to as everyonealso thinks he is stupid.That's something that is overly obvious but rarely, if ever, stated.Pmb === > Awwww... ain't that cute? The two little cranks are playing together!> When I was working at MIT, a professor of my acquaintance (I'll say> The idea was to be kind, and yet at the same time, not waste his own> time which was better spent on other things. He thought that just> ignoring people or dismissing them would be unkind. And since he> worked at the AI lab and was high pro?e in various ways, he gets> look at it, see what the topic was, and pull the ?e of other letters> about the same topic. Then he would reply with a form letter saying> something like: I'm sorry, but I don't have the expertise to address> your letter. The following people may be better suited to discuss> your points. Then he would add the names and addresses of a few of> Now this struck me as wonderfully brilliant. Simultaneously being> kind, moreover, giving the nut someone to talk to who will listen and> respond in the same vein, satisfying both, keeping his own time free> for other things--amazing.> Have you ever noticed that cranks and nutters are harder to> get together than Pandas during mating season? If you watch> them (the cranks) frolic on usenet, you'll notice that they> seem to, for the most part, utterly ignore each other. It's> as though they instinctively know to avoid their own species.Fallacy of af?ming the consequent. Crank, your post is self-contradictory.JS === Most texts include both ?ite and in?ite (countably in?ite)sets under the countable de?ition. A few (Rudin, e.g.) de?ecountable sets as countably in?ite only. The latter de?itionappears to create less confusion. For example, under the former theexpression: oo U (A_j) j=1 is taken to mean the countable union of A_j's whether {A_j} isin?ite or ?ite, with oo always present. Can someone pleaseindicate what the bene?s/drawbacks of each are, as they apply todevelopment of the relevant theory of such ?lds as probability,topology, etc. Many thanks. === > Most texts include both ?ite and in?ite (countably in?ite)> sets under the countable de?ition. A few (Rudin, e.g.) de?e> countable sets as countably in?ite only. The latter de?ition> appears to create less confusion. For example, under the former the> expression:> oo> U (A_j)> j=1> is taken to mean the countable union of A_j's whether {A_j} is> in?ite or ?ite, with oo always present. Can someone please> indicate what the bene?s/drawbacks of each are>Tho not much used, some make distinction'denumerable' for ?in?itely countable'.If countable means denumerable, then uncountable would mean?ite or having cardinality greater than the integers.Given a sequence a_j, the set { a_j | j in N } is countable. === > Most texts include both ?ite and in?ite (countably in?ite)> sets under the countable de?ition. A few (Rudin, e.g.) de?e> countable sets as countably in?ite only. The latter de?ition> appears to create less confusion. For example, under the former the> expression:> oo> U (A_j)> j=1> is taken to mean the countable union of A_j's whether {A_j} is> in?ite or ?ite, with oo always present. Can someone please> indicate what the bene?s/drawbacks of each are> Tho not much used, some make distinction> ?denumerable' for ?in?itely countable'.> If countable means denumerable, then uncountable would mean> ?ite or having cardinality greater than the integers.> Given a sequence a_j, the set { a_j | j in N } is countable.I ?d that I am reluctant to label a set like {1,2,3} uncountable. === > Let's see. > do). Then, by the de?ition of the GCD, s(m) divides> both g_1(m) and f. Let t(m) = f/s(m) and c(m) = g_1(m)/s(m).> With that test of your imagination above, you now have that t(m) = f.> If f divides g_1*g_2 and f is coprime to g_1 then f divides g_2.> Duh!> Not necessarily. You see it keeps coming back to the same thing,> which is your apparent inability to comprehend a certain possibility.> Ok, now then, can you comprehend that you're now *assuming* the very> thing that supposedly you were trying to prove?> James HarrisTheorem:In any commutative ring with identity, (A,+,*), if(1) f divides a*b (i.e., f*g = a*b, for some g in A), and (2) f and a are coprime (i.e., f*u + a*v = 1, for some u,v in A),then f*w = b for some w in A.Proof: (1) a*b = f*g given (2) (a*v)*b = f*g*v v times both sides of (1) (3) (1 - f*u)*b = f*g*v) Substituting a*v = 1- f*2 (4) b - f*u*b = f*g*v distributing on LHS (5) b = f*u*b + f*g*v shifting terms (6) b = f*(u*b + g*v) factoring on RHS (7) b = f*w where w = u*b + g*v, in A Q.E.D.Who is assuming what, now?As the previous poster says, Duh! === [I changed the labels of the hypothesis from (1) and (2) to (i) and (ii), to avoid confusion with the labels of hte steps in the proof.]>Theorem:>In any commutative ring with identity, (A,+,*), if> (i) f divides a*b (i.e., f*g = a*b, for some g in A), and >(ii) f and a are coprime (i.e., f*u + a*v = 1, for some u,v in A),>then f*w = b for some w in A.>Proof:> (1) a*b = f*g given> (2) (a*v)*b = f*g*v v times both sides of (1)> (3) (1 - f*u)*b = f*g*v) Substituting a*v = 1- f*2 > (4) b - f*u*b = f*g*v distributing on LHS> (5) b = f*u*b + f*g*v shifting terms> (6) b = f*(u*b + g*v) factoring on RHS> (7) b = f*w where w = u*b + g*v, in A> Q.E.D.>Who is assuming what, now?The dif?ulty here is on the parenthetical comment on (ii): thestandard use in algebraic number theory (though not necessarilyelsewhere, as I have been often reminded by Bill Dubuque) is thatcoprime for algebraic integers means exactly what the parentheticalcomment states: that there is a linear combination equal to 1; just asin the integers.However, James has repeatedly stated that he uses coprime to meanthe only common factors are units.The two statements are not equivalent in general, though they areequivalent in any PID, or more generally in any Bezout Domain.Let me label them for ease:Let R be an integral domain, x and y in R; we say that x and y arecoprime in R if and only if: (A) there exist r, s, in R such that rx+sy = 1. (B) if u in R divides both x and y (in R), then u is a unit in R.It is easy to verify that (A)->(B). The harder part is to show that(B)->(A) for the algebraic integers (it is not true in general; e.g.,in Z[sqrt(-5)], the standard example, 2 and 1+sqrt(-5) are coprime inthe sense of (B), but not in the sense of (A)).The only proof I know for (B)->(A) in the algebraic integers uses bothunique factorization into prime ideals, and more importantly the?iteness of the class number, which is of course a fairly dif?ultresult. In The Theory of Algebraic Integers, by Richard Dedekind(translated by John Stillwell, Cambridge University Library, CambridgeUniversity Press 1996), Dedekind states in Chapter 3, Generalproperties of algebraic integers, section 14 divisibility o?tegers, ?al paragraph (pp. 106 in my edition): A deeper investigation will enable us to see that two nonzero [algebraic] integers a and b have a ->greatest common divisor<- [emphasis in the original], which can be put in the form aa'+bb', where a' and b' are [algebraic] integers. This important theorem is NOT AT ALL EASY TO PROVE [emphasis added] with the help of the principles developed thus far, but we shall later (section 30) be able to derive it very simply from the theory of ideals.The proof does indeed appear in Section 30, the ?al section of thework. Here's a paraphrase, which uses ?iteness of the class number. THEOREM. Any two algebraic integers a, b have a common divisor d which can be expressed in the form d = aa' + bb', where a' and b' are likewise algebraic integers.Note that the fact that it is a common divisor AND a linearcombination implies that it is a greatest common divisor in theusual sense; for if e is any other common divisor of a and b, then edivides aa'+bb', hence divides d. Proof of Theorem. We may assume both a and b are nonzero. Let K be anumber ?ld containing both a and b (say, K= Q(a,b)), and let A beits ring of integers. Let h be the class number of A. Factoring theideals (a) and (b) into prime ideals, we have(a) = A*D; (b) = B*Dwhere A, B, D are ideals, products of prime ideals, A and B arerelatively prime, and D is the gcd of (a) and (b). Since h is theclass number, D^h is principal, say D^h = (e), with e in A.Since (a^h) = (a)^h and (b^h) = (b)^h are divisible by D^h = (e), itfollows that both a^h and b^h are in (e), hence there exist r and s inA such thata^h = e*r, b^h = e*s;by unique factorization into primes, (r) = A^h and (s) = B^h. Since Aand B are relatively prime, so are (r) and (s). Therefore, thesmallest ideal containing both r and s is A itself. That is, (r,s)=A;since 1 is in A, there exist elements u and v in A such that r*u + s*v = 1. Multiplying through by e, we havee = e*1 = e*r*u + e*s*v = a^h*u + b^h*vLet d = e^{1/h}, which is an algebraic integer (being the root of x^h - e, a monic polynomial with algebraic integercoef?ients). Since e is a common divisor of a^h and b^h, d is acommon divisor of a and b; and since h>0, we may writeu*a^{h-1} = a'*d^{h-1},v*a^{h-1} = b'*d^{h-1}for some algebraic integers a' and b'; namely, if a = d*x, then a' =u*x^{h-1}, and if b = d*y, then b' = v*y^{h-1}aa' + bb' = a*u*x^{h-1} + b*v*y^{h-1} = d*x*u*x^{h-1} + d*y*v*y^{h-1} = d*u*x^h + d*v*y^h = d(u*x^h + v*y^h)But a=d*x implies a^h = d^h*x^h = e*x^hand b=d*y implies b^h = d^h*y^h = e*y^h.We know that e = a^h*u + b^h*v, from which we havee = a^h*u + b^h*v = e*x^h*u + e*y^h*v = e(x^h*u + y^h*v)Since a and b are nonzero, e is nonzero, so we conclude that x^h*u + y^h*v = 1. Therefore,aa' + bb' = d(u*x^h + v*y^h) = d(1) = d.We already know that d is a common divisor of a and b; and now we haveshown it is an algebraic integer linear combination of a and b, asclaimed. QED-- === =========================================== === ====It's not denial. I'm just very selective about what I accept as reality. Calvin (Calvin and Hobbes) === ====================================== === =========Arturo Magidinmagidin@math.berkeley.edu===> [I changed the labels of the hypothesis from (1) and (2) to (i) and> (ii), to avoid confusion with the labels of hte steps in the proof.]>Theorem:>In any commutative ring with identity, (A,+,*), if> (i) f divides a*b (i.e., f*g = a*b, for some g in A), and>(ii) f and a are coprime (i.e., f*u + a*v = 1, for some u,v in A),>then f*w = b for some w in A.>Proof:> (1) a*b = f*g given> (2) (a*v)*b = f*g*v v times both sides of (1)> (3) (1 - f*u)*b = f*g*v) Substituting a*v = 1- f*2> (4) b - f*u*b = f*g*v distributing on LHS> (5) b = f*u*b + f*g*v shifting terms> (6) b = f*(u*b + g*v) factoring on RHS> (7) b = f*w where w = u*b + g*v, in A> Q.E.D.>Who is assuming what, now?> The dif?ulty here is on the parenthetical comment on (ii): the> standard use in algebraic number theory (though not necessarily> elsewhere, as I have been often reminded by Bill Dubuque) is that> coprime for algebraic integers means exactly what the parenthetical> comment states: that there is a linear combination equal to 1; just as> in the integers.> However, James has repeatedly stated that he uses coprime to mean> the only common factors are units.> The two statements are not equivalent in general, though they are> equivalent in any PID, or more generally in any Bezout Domain.> Let me label them for ease:> Let R be an integral domain, x and y in R; we say that x and y are> coprime in R if and only if:> (A) there exist r, s, in R such that rx+sy = 1.> (B) if u in R divides both x and y (in R), then u is a unit in R.> It is easy to verify that (A)->(B). The harder part is to show that> (B)->(A) for the algebraic integers (it is not true in general; e.g.,> in Z[sqrt(-5)], the standard example, 2 and 1+sqrt(-5) are coprime in> the sense of (B), but not in the sense of (A)).> The only proof I know for (B)->(A) in the algebraic integers uses both> unique factorization into prime ideals, and more importantly the> ?iteness of the class number, which is of course a fairly dif?ult> result. In The Theory of Algebraic Integers, by Richard Dedekind> (translated by John Stillwell, Cambridge University Library, Cambridge> University Press 1996), Dedekind states in Chapter 3, General> properties of algebraic integers, section 14 divisibility of> integers, ?al paragraph (pp. 106 in my edition):> A deeper investigation will enable us to see that two nonzero> [algebraic] integers a and b have a ->greatest common divisor<-> [emphasis in the original], which can be put in the form aa'+bb',> where a' and b' are [algebraic] integers. This important theorem is> NOT AT ALL EASY TO PROVE [emphasis added] with the help of the> principles developed thus far, but we shall later (section 30) be> able to derive it very simply from the theory of ideals.> The proof does indeed appear in Section 30, the ?al section of the> work. Here's a paraphrase, which uses ?iteness of the class number.> THEOREM. Any two algebraic integers a, b have a common divisor d> which can be expressed in the form d = aa' + bb', where a' and b' are> likewise algebraic integers.> Note that the fact that it is a common divisor AND a linear> combination implies that it is a greatest common divisor in the> usual sense; for if e is any other common divisor of a and b, then e> divides aa'+bb', hence divides d.> Proof of Theorem. We may assume both a and b are nonzero. Let K be a> number ?ld containing both a and b (say, K= Q(a,b)), and let A be> its ring of integers. Let h be the class number of A. Factoring the> ideals (a) and (b) into prime ideals, we have> (a) = A*D; (b) = B*D> where A, B, D are ideals, products of prime ideals, A and B are> relatively prime, and D is the gcd of (a) and (b). Since h is the> class number, D^h is principal, say D^h = (e), with e in A.> Since (a^h) = (a)^h and (b^h) = (b)^h are divisible by D^h = (e), it> follows that both a^h and b^h are in (e), hence there exist r and s in> A such that> a^h = e*r, b^h = e*s;> by unique factorization into primes, (r) = A^h and (s) = B^h. Since A> and B are relatively prime, so are (r) and (s). Therefore, the> smallest ideal containing both r and s is A itself. That is, (r,s)=A;> since 1 is in A, there exist elements u and v in A such that> r*u + s*v = 1.> Multiplying through by e, we have> e = e*1 = e*r*u + e*s*v> = a^h*u + b^h*v> Let d = e^{1/h}, which is an algebraic integer (being the root of> x^h - e, a monic polynomial with algebraic integer> coef?ients). Since e is a common divisor of a^h and b^h, d is a> common divisor of a and b; and since h>0, we may write> u*a^{h-1} = a'*d^{h-1},> v*a^{h-1} = b'*d^{h-1}> for some algebraic integers a' and b'; namely, if a = d*x, then a' => u*x^{h-1}, and if b = d*y, then b' = v*y^{h-1}> aa' + bb' = a*u*x^{h-1} + b*v*y^{h-1}> = d*x*u*x^{h-1} + d*y*v*y^{h-1}> = d*u*x^h + d*v*y^h> = d(u*x^h + v*y^h)> But a=d*x implies a^h = d^h*x^h = e*x^h> and b=d*y implies b^h = d^h*y^h = e*y^h.> We know that> e = a^h*u + b^h*v, from which we have> e = a^h*u + b^h*v> = e*x^h*u + e*y^h*v> = e(x^h*u + y^h*v)> Since a and b are nonzero, e is nonzero, so we conclude that> x^h*u + y^h*v = 1. Therefore,> aa' + bb' = d(u*x^h + v*y^h)> = d(1)> = d.> We already know that d is a common divisor of a and b; and now we have> shown it is an algebraic integer linear combination of a and b, as> claimed. QED>Rats. Now we're back to ?pages of math', an environment over? withopportunities for JSH to misinterpret and twist the facts. Virgil'soriginal proof was so simple that I was sure even JSH couldn't escape it.Oh well.l8r, Mike N. Christoff === > Let's see.> do). Then, by the de?ition of the GCD, s(m) divides> both g_1(m) and f. Let t(m) = f/s(m) and c(m) = g_1(m)/s(m).> With that test of your imagination above, you now have that t(m) =f.> If f divides g_1*g_2 and f is coprime to g_1 then f divides g_2.> Duh!> Not necessarily. You see it keeps coming back to the same thing,> which is your apparent inability to comprehend a certain possibility.> Ok, now then, can you comprehend that you're now *assuming* the very> thing that supposedly you were trying to prove?> James Harris> Theorem:> In any commutative ring with identity, (A,+,*), if> (1) f divides a*b (i.e., f*g = a*b, for some g in A), and> (2) f and a are coprime (i.e., f*u + a*v = 1, for some u,v in A),> then f*w = b for some w in A.> Proof:> (1) a*b = f*g given> (2) (a*v)*b = f*g*v v times both sides of (1)> (3) (1 - f*u)*b =